# Exponential Regression using a Linear Model

Sometimes linear regression can be used with relationships which are not inherently linear, but can be made to be linear after a transformation. In particular, we consider the following exponential model:

Taking the natural log (see Exponentials and Logs) of both sides of the equation, we have the following equivalent equation:

This equation has the form of a linear regression model (where I have added an error term ε):

Observation: Since αeβ(x+1) = αeβx · eβ, we note that an increase in x of 1 unit results in y being multiplied by eβ.

Observation: A model of the form ln y = βx + δ is referred to as a log-level regression model. Clearly any such model can be expressed as an exponential regression model of form y = αeβx by setting α = eδ.

Example 1: Determine whether the data on the left side of Figure 1 fits with an exponential model.

Figure 1 – Data for Example 1 and log transform

The table on the right side of Figure 1 shows ln y (the natural log of y) instead of y. We now use the Regression data analysis tool to model the relationship between ln y and x.

Figure 2 – Regression data analysis for x vs. ln y from Example 1

The table in Figure 2 shows that the model is a good fit and the relationship between ln y and x is given by

Applying e to both sides of the equation yields

We can also see the relationship between  and  by creating a scatter chart for the original data and choosing Layout > Analysis|Trendline in Excel and then selecting the Exponential Trendline option. We can also create a chart showing the relationship between  and ln  and use Linear Trendline to show the linear regression line (see Figure 3).

Figure 3 – Trend lines for Example 1

As usual we can use the formula y = 14.05∙(1.016)x described above for prediction. Thus if we want the y value corresponding to x = 26, using the above model we get ŷ  =14.05∙(1.016)26 = 21.35.

We can get the same result using Excel’s GROWTH function, as described below.

Excel Functions: Excel supplies two functions for exponential regression, namely GROWTH and LOGEST.

LOGEST is the exponential counterpart to the linear regression function LINEST described in Testing the Slope of the Regression Line. Once again you need to highlight a 5 × 2 area and enter the array function =LOGEST(R1, R2, TRUE, TRUE), where R1 = the array of observed values for y (not ln y) and R2 is the array of observed values for x, and then press Ctrl-Shft-Enter. LOGEST doesn’t supply any labels and so you will need to enter these manually.

Essentially LOGEST is simply LINEST using the mapping described above for transforming an exponent model into a linear model. For Example 1 the output for LOGEST(B6:B16, A6:A16, TRUE, TRUE) is as in Figure 4.

Figure 4 – LOGEST output for data in Example 1

GROWTH is the exponential counterpart to the linear regression function TREND described in Method of Least Squares. For R1 = the array containing the y values of the observed data and R2 = the array containing the x values of the observed data, GROWTH(R1, R2, x) = EXP(a) * EXP(b)^x where EXP(a) and EXP(b) are as defined from the LOGEST output described above (or alternatively from the Regression data analysis). E.g., based on the data from Example 1, we have:

GROWTH(B6:B16, A6:A16, 26) = 21.35

which is the same result we obtained earlier using the Regression data analysis tool.

GROWTH can also be used to predict more than one value. In this case, GROWTH(R1, R2, R3) is an array function where R1 and R2 are as described above and R3 is an array of x values. The function returns an array of predicted  values for the x values in R3 based on the model determined by the values in R1 and R2.

Observation: Note that GROWTH(R1, R2, R3) = EXP(TREND(R1, R2, LN(R3)))

### 92 Responses to Exponential Regression using a Linear Model

1. Kevin Urben says:

You have made an error when you take the log of both sides of the equation, log(a+b) is not equal to log(a)+log(b).

• Charles says:

Hi Kevin,
You are 100% correct. I have now removed this unnecessary and careless mistake from the website. Thanks for finding the error.
Charles

• kyaezin says:

Y=aexp^(-x/b)
can any one solve this equation?

• Charles says:

Take the natural log of both sides of the equation and then use properties of logs and exp:

ln y = ln(a exp^(-x/b))
ln y = ln a + ln(exp^(-x/b))
ln y = ln a – x/b
x/b = ln a – ln y
x = b ln (a/y)

Charles

2. gary says:

hello,
this is a semi-log regression right?
how do we create a log-log regression?

• Charles says:

Gary,
What I call exponential regression is sometimes called log-level regression. I have just updated the Exponential Regression webpage to say this. In the next day or so I will be adding a new webpage called Power Regression. This is the same as log-log regression and on this webpage I will explain how to perform this type of regression in Excel. Thanks for your question.
Charles

Update (28 Oct 2013): The changes referenced above have now been made. Please check out the new Power Regression and Multiple Regression with Logarithmic Transformations webpages.

3. Kirsten says:

Hello,
I am doing a y=ln(x) regression however I also need to use the residuals. Am I able to just plot the residuals against the original x values? The residuals are in same units as the y values which have not been transformed, so the residuals should still be true for the original data. Is this correct?

• Charles says:

Kirsten,

The residuals are based on the model used, not really the original data. The residuals are the difference between the observed and predicted values of the y’s. E.g. suppose the observed values of x are 7.4, 1.6, 2.5, 4.5, 12.2, 8.2, 5.5, 2.1, 2.9 with corresponding observed y values 5.2, 1.9, 2.9, 3.5, 6.7, 5.5, 4.2, 2.1, 3.1. It turns out that the regression model y=ln(x) is a pretty good fit for the data (R-square = .969). The residuals for these data points are 0.006686154, 0.268702213, 0.230696105, -0.536419692, 0.343853242, 0.067925862, -0.303153788, -0.163780455, 0.08549036. You can then plot these values against x. You will see a pretty random plot.

Charles

• Jam One says:

Exactly how do you arrive at your residual values? Can you please explicitly show us the calculation? As an example, the residual between the observed value x of 7.4 and the observed value y of 5.2.

• Charles says:

If you are modelling y = a*e^(bx), then the linear model is lny = bx + lna. The residual of the linear model is the difference between the observed value of lny and the predicted value of lny. For the model in Example 1 of the referenced webpage, when x = 7.4 then the model predicts y to to be 15.828 (although caution should be used since the value for x is a bit output the observed data range). Since you said that the observed value of y is 5.2, the observed value of ln y is ln 5.2 = 2.001 and the predicted value of ln y is ln 15.828 = 2.762, and so the residual is 2.001 – 2.762 = 0.761.

Technically the residual for the exponential model is simply 5.2 – 15.828 = -10.628, although the value in the above paragraph may be more relevant for many purposes.

Charles

4. R. Andre says:

correction:

I meant to say

Is it correct to say that if x increases by delta_x units, y increases by 100*(eˆ(b1*delta_x)-1)%? (for b1 much larger that 0.1)

• Charles says:

Hi,
Yes this is correct. I believe this is true for any value of b1, not just for b1 much larger than 0.1, using simple algebra.
Charles

5. Jorj McKie says:

For exponential, logarithmic and power trend fits, Excel uses the least square method on the data pairs [x, ln(y)] (in the exponential case).

From this approach inherit two issues:

1) The R-squared given in charts is the one of the linear fit to those [x, ln(y)] pairs. NOT the R-squared of your original data! So do not rely on this value in the chart!
This fact is documented somewhere in Excel … not too easy to find though.

2) The overall approach will in general NOT deliver an optimal fit!!! I can provide examples, where the Excel trend (no matter if calculated as a chart trendline or by a worksheet function like GROWTH) is worse than an exponential fit calculated e.g. with an Levenberg-Marquardt algorithm.
This fact is NOT documented in Excel – maybe not even known to Microsoft.
On the contrary, Excel documentation states that its trends are “best fit”. They are not – always.

So, if you want to be sure whether your data follow an exponential, logarithmic or power pattern: do not rely on Excel!

• Charles says:

Jorj,
Sorry that I haven’t responded to your comment earlier, but I have been on vacation for the past few weeks. I appreciate your serious comment and plan to look into it. Based on your comments I may need to provide a new version of the referenced regression algorithms that provide a better fit along the lines that you have suggested.
Charles

• Hi Charles.

I just came across this thread and if you have already addressed this issue elsewhere, just ignore my post.

We can view the exponential model as follows. Y has a log-normal distribution while Ln(Y) has a normal distribution. The relationship between the Normal and Log-normal distribution is well defined. For instance, if the variable Ln(Y) has a Normal distribution with mean = mu and standard deviation = sigma, then the variable Y has Log-normal distribution (with parameters mu and sigma). The mean of Y is given by e^(mu + 0.5*Sigma^2). It is NOT e^(mu). Similarly higher order moments can be defined (see Wikipedia Log-normal).

The following notation is not exactly right, but I hope it conveys the message. Excel predicted values are off because of the fact that Excel trend line does prediction as e^(bo + b1X) (where b0 and b1 are the result of regressing Ln(Y) on X). But e^(bo + b1X) is not the expected value of Y given X. The correct expression for the expected value of Y given X should be e^(b0 + b1X + 0.5*MSE). So Excel will consistently under-estimate the expected value of Y given X (unless it is a perfect fit in which case MSE will be 0).

This would be the solution for the Exponential model, but not necessarily for the other models. So Jorj is correct in that the Excel may not be the best approach for non-linear estimation. Most people assume that transformations are easy … they usually are not. And this is a case in point.

• Charles says:

Krish,
Thanks for your comments. I had been reluctant to spend the time necessary to implement the Levenberg-Marquardt algorithm as suggested by Jorj, but I can see that it is not sufficient to simply accept the approach used by Excel. Shortly I will modify the website to at least comment on the discrepancy and try to come up with a compromise solution.
Charles

• Charles says:

Krish,
The latest release of the software, Release 3.8, provides a nonlinear regression solution to the exponential model.
Charles

• Charles says:

Jorj,
The latest release of the software, Release 3.8, provides a nonlinear regression solution to the exponential model. I expect to add a description of how to use these new capabilities later today.
Charles

6. petr says:

Hey, I have a doubt. If we have a data and we need to find the relation between them,we use correl to see if they have any linear relation between them. Likewise is there any counterpart for correl to see the exponential relation between the data. If we have a data,how can we come to a conclusion that they are exponentially related and then use logest or growth to predict the further values.please help me

• Charles says:

Petr,
As described on the referenced webpage, if x and y have a exponential relationship, i.e. y = abe^x, then ln y = bx + ln a, which is a lineaar relationship. Thus you could use correl between ln y and x to test whether x and y have an exponential relationship.
Charles

• kudzy says:

wise contributions from all of you guys, my problem has been solved. thank you

7. jaimin pandya says:

Y=αxe^βx

can any one solve this example???
i cant fit into Y= mX + C

• Charles says:

By “solve” do you mean put y=αxe^βx into the form Y = mX + C? This doesn’t seem likely since logy = βx + logα + logx takes the form Y = mX + logX + C.
Charles

8. Pyay Zaw Htet says:

I would like to know about decline curves and rates.
y = 2E+21e-0.002x

I want to know decline rate is -0.02%? My problems are to know rates and -0.002.

• Charles says:

.002 is the exponential decay constant, which results in a decline rate of .2%.
Charles

9. Taniya says:

Can you provide any real life example of log-linear model??

• Charles says:

Taniya,
Take any example you have for the chi-square test for independence (of two variables) and simply add another variable. Log-linear models analyze the resulting 3-way contingency tables.
Charles

Pls I need a solution to this problem where can I get semi-log regression and double log regression in SPSS or which software can I use to solve it

• Charles says:

Sorry, but I am not that familiar with SPSS. I do my statistical analysis using Excel and my software, the Real Statistics Resource Pack.
Charles

11. Basha says:

how all functions (linear, semi log, double log and exponential) can be applied

• Charles says:

Sorry, but I am not sure that I understand your question, although perhaps the following webpage provides the information you are looking for.
Power Regression
Charles

12. Alyssa says:

In order to run the regression why don’t you have to take the natural log form of both x and y before coming up with the equation? Then once you have it in the natural log form would you just take the equation and set it equal to marginal cost to find the profit maximizing quantity? Thank you.

• Charles says:

Alyssa,

I have taken the natural log of both sides of the equation. Remember that LN(EXP(x)) is x.

Regarding “…marginal cost to find the profit maximizing quantity”, you apparently have a particular application of exponential regression in mind, but exponential regression can be used to address a variety of applications, not just those from economics.

Charles

13. Jai Desai says:

What if we take log to the base 10 i.e. normal log instead of ln i.e. natural log?

• Charles says:

You can use log base 10 instead of the natural log, but the answers will differ by a constant factor since LN(x) = a Log(x) for any x where a = 2.302585…
Charles

14. Jai Desai says:

15. Pingback: Linest and the if function

16. Moshe Khan says:

Dear Charles

It might be possible that I am not doing it properly, because I still can not download it to my computer and have it as an Add in tool
Thanks
MK

• Charles says:

Moshe,

To download the software, just go to the following webpage and click on the Free Download button (assuming that you are using Excel 2010, 2013 or 2016 for Windows)
Real Statistics Resource Pack

To install the software after you have downloaded it, perform the following steps:

1. Open Excel, but don’t try to open the realstats.xlam file that you downloaded
2. Press Alt-TI (i.e. hold the Alt key down and simultaneously press T followed by I)
3. On the Add-Ins dialog box that appears press the Browse button and locate where you stored the realstats.xlam file that you downloaded (this done in a similar manner as when you Open a file)
4. Once you have done this, make sure that the Realstats option on the Add-Ins dialog box is checked and click the OK button

17. Rachel says:

Under what circumstances would it be appropriate to log transform only the independent variable for an exponential regression?

Would that be considered a log-level or a different type?

• Charles says:

Rachel,
This transformation is appropriate when it provides a better fit for your data.
This is level-log regression as described on the webpage http://www.real-statistics.com/regression/power-regression/.
Charles

• Rachel Rotz says:

Thanks Charles. These pages are very helpful.

However, a power regression includes the transformation of both the x and y axis.

log-log: power
log-level: exponential

What is it called when you just transform the independent variable?

• Charles says:

Rachel,
See Power Regression
Charles

• leo onix says:

thank you, i will try to understand this stuff. I actually a bit confused. 🙁

18. Zaynab says:

When we use other functional forms to run the regression analysis, why do we then choose the functional form with the least MS residual to analyse the result?

• Charles says:

Zaynab,
I don’t see any reference to least MS residual on the referenced webpage. In general, the logic behind minimizing the squared error is as described on the webpage
Maximum Likelihood Function.
Charles

19. Ahmed Salah says:

Hi,
I saw in some papers that the coefficients are interpreted as semi-elasticity without outlining the initial model. Does that mean that the model is in log-level form ?

• Charles says:

Ahmed,
Yes. This is explained on the following webpage>
Power Regression
Charles

20. Nazia says:

Hi,
Can you please elaborate, if we have an exponentially decaying function, when we need to/ we can use linear fiitting for interpolation instaed of using exponentially decaying curve?

• Charles says:

Nazia,
As described on the referenced webpage, you can always model data that fits an exponential function using a linear model. The non-linear model (as described on the Real Statistics website) will be a little more accurate.
Charles

21. Lily says:

Hello,

Can you please explain to me how you would do the regression data analysis…? I want to know how to get the values in figure 2, Thank you!

22. Dennis says:

thanks for the wonderful post; please check out this question how do you solve this
Y=aBexp-kt

• Charles says:

Dennis,
If y = aexp(-kx), then lny = -kx + lna, which takes the form of a linear regression y’ = -kx + a’ where y’ = lny and a’ = lna.
Charles

23. Dennis says:

thanks for the post
Y=αe^-βt

can any one solve this example???
i can fit into Y= α+βt
but answer … EInY=-0.37182, E(InI)^2=3.45846, EtInI=-1.37554
NOTE E MEANS SUM

24. Pranav says:

Hi Charles,

How to determine the vlaues of a and b without using excel functions ( on paper)

Thanks

• Pranav says:

Can linear model intepretation of slope and intercept be applied directly after take the natural log ?

• Charles says:

Pranav,
Yes, as described on the referenced webpage.
Charles

• Charles says:

Pranav,
See the following webpage for the calculation of a and b (for the linear case) by hand.
http://www.real-statistics.com/regression/least-squares-method/
Charles

• Pranav says:

Very detailed information. Thanks for the link.

25. Pranav says:

I have further question after finding the intercepts.
Using Y=a.exp(kt)
In order to determine the value of t for given max/min value of Y (max = 100, min = 0 in my application.) wouldn’t the value of t be Inf. ?
By subsitution, t = (ln(Y)-ln(a) )/ k

Also , When the logarithmic distance between the 2 points to determine the constants is ‘more’ or ‘less’ , how would be the prediction of t ?

• Charles says:

Pranav,
Since t = (ln(Y)-ln(a))/ k, t would be undefined when y = 0, but would have a finite value when y = 100.
Sorry, but I don’t understand your other question.
Charles

• Pranav says:

Sorry about not posing the question clearly.

If we had to calculate the value of t at end positions(ex:1 and 100),
how would the separation between the 2 points used to determine the values of a and k affect the value of t.

• Charles says:

Pranav,
Sorry, but I still don’t understand the question.
Charles

26. Brittany says:

Hi,
What do you suppose is the simplest way in which I could show and explain how the method of obtaining an exponential function via a linear model is valid to all exponential functions?
I have already demonstrated that an exponential function (ab^x) can be found by first obtaining a linear function using using logs and graphing however need to further prove that it can be applied to all exponential functions.
Would the use of natural log be useful for the (ae^kx)? If so how would I go about showing this with a given set of data? My first and second x and y values are (0,100) and (1,62).

• Charles says:

Brittany,
Yes, the same technique should apply. y = ae^(kx) means that lny = lna + kx.
Charles

27. Alan says:

Hi,
Does anyone know if there is a way to calculate the whole thing by hand? The whole formula?

• Charles says:

Alan,
The referenced webpage tells you how to perform exponential regression based on linear regression. If you look at the webpages on linear regression as well, you will see how to perform linear regression (and therefore exponential regression) by hand. This will not consist of a single formula.
Charles

28. yousif says:

Q. what is the suitable statistical method to calculate and measure the time once we knew the value of y at t0 and value of y at tn ?
In the graph below

y

t
t0 tn
tn-t0=?

Q. what is the suitable statistical method to calculate and measure the time once we knew the value of y at t0 and value of y at tn ?

I think we calculate the time using linear interpolation and certain numerical methods.
In this case, I need to know the statistical method to estimate the time. And how?
We can rewrite the question as follows:
How long time the process takes from initial value y0 to end value yn ?

• Charles says:

Yousif,
Sorry, but I don’t understand your question.
Charles

29. Can you provide any real life example of log-linear model??

• Charles says:

Examples can be found in Howell’s textbook (see Bibliography). Other real examples can be found in various textbooks and online.
Charles

30. Zulfiqar says:

Hi,
I am in a bit puzzled by following question. I need linear regression to model the following form: y = Ae^(-0.3t); as you can see that b is already known in this case. So, how to do this kind of regression in excel. Actually, I want to compare results with my hand calculations. I have done hand calculations using two methods:
1. without data transformation
2. with data transformation
Now, using these two approaches, I get slightly different values for A. The value for A obtained by using model without data transformation gives a smaller sum of squared errors, as compared to using model with data transformation. I am puzzled by this variation as my understanding is that both systems should give same answers. Is my understanding correct? Am I missing something?

• Charles says:

This probably depends on the nature of the data transformation. There is no reason to assume that you will get the same answer from options 1 and 2.
Charles

• Zulfiqar says:

Thanks, for the clarification. Can you please also guide some source where we can learn about scaled residuls.

Thanks,
Zulfiqar

31. Tina says:

Hello, I am new in the field of statistics, I would like to ask how to calculate the growth rate with time series data.I wanna know the crop yield growth rate and calculate the instability in growth.
I have this exponential equation:
Y=ab^t
Where Y=yield
a=constant
b=trend
t=time(from 2000to 2014)
I have the data of crop Yield for this period.
The formula for instability index :CV*(1-R squared)square root

• Charles says:

Tina,
Have you tried to apply the approach on the referenced webpage?
Charles

• Tina says:

@Charles, Yes,I have tried many in this way:

1)calculate ln y
2)regress ln y x
3) calculate exp(b)
When I calculate the Compound Annual Growth rate from the value
of b obtained(CAGR=(antilog b-1 )*100,i find the answer different when i use the common formula to calculate CAGR in excel.

• Tina says:

@Charles, can you refer where i can find how to calculate Compound Annual Growth rate with empirical study(not the common formula in excel )

Thanks

• Charles says:

Tina,
If all your data lies on the regression line then the CAGR should agree with the usual value. If not, then you could expect the CAGR not to be the same as that generated by the usual formula, although if the data lies close to the regression line, then the results should be similar.
Charles

32. MOHAIMINUL HAQUE says:

hi will i able to use this exponential equation for forecasting?

• Charles says:

Yes

• vinodh nallasamy says:

Is there any Example or link for that. i want manual calculation

• Charles says:

Vinodh,
This webpage shows how to transform an exponential regression model into a linear regression model. The webpage uses Excel’s Regression data analysis tool to carry out the regression. If you want to do this part manually, then please see the following webpage:
http://www.real-statistics.com/regression/least-squares-method/
Charles

33. Mallika says:

Let’s say we have y = a exp(b*x) + e1 where e1 is the error term. Then we solve ln y = lna + b ln x + e2 where e2 is the error term using the regression. What is the relationship between e1 and e2 so that linearization is accurate?

• Charles says:

Mallika,
e2 = ln(a*exp(b*x)+e1) – (lna + b*lnx)
Charles

34. mitku says:

hello guys ,i want to regress a given data points of t and y ,then find the coefficients and constants for the following models .
y=a*exp(bt^c)
y=a*exp((bt)^c)
can any one help me how to find the a,b and c values by regression for a given datas of experiment of t and y?

• Charles says:

Mitku,
If we take the log of both sides of the first equation we get ln y = ln a + bt^c. Now let y’ = ln y, a’ = ln a and x = t^c, then the equation takes the form
y’ = bx + a’. You can then use ordinary regression to find the values for a’ and b. Then a = exp(a’). The only problem with this approach is that c is treated as a constant. If you want to calculate a value for c, then this approach won’t work and you will need to use a non-linear regression approach, e.g. using Solver as I have shown on the website for exponential regression. The situation is similar for the second equation.
Charles