# Microsoft Knowledge Base

## Excel: Finding an Exponential Curve Using Natural Log

Last reviewed: September 12, 1996 Article ID: Q104449
The information in this article applies to:

- Microsoft Excel for Windows, versions 2.x, 3.0, 4.0, 4.0a
- Microsoft Excel for OS/2, version 2.2, 3.0
- Microsoft Excel for the Macintosh, version 2.x, 3.0, 4.0

## SUMMARY

In Microsoft Excel, the LOGEST() function calculates an exponential curve that fits data to the equation y=b*m^x and returns b and m for the given data.

If you want to use a natural logarithmic relation in the form of y=b*e^(n*x), then n should be taken to be equal to LN(m), where m is the value returned by LOGEST().

## MORE INFORMATION

The LOGEST() function returns:

       y=b*m^x

However, if you want LOGEST() return the natural log form, then the formula should be:

       y=b*e^(n*x)

The above is algebraically derived as follows:

       e^(n*x)=(e^n)^x

By substituting the above into e^(n*x)=(e^n)^x, it results in the following:

       y=b*(e^n)^x.

LOGEST() returns b and m. By comparing the LOGEST() form y=b*m^x with the natural log form y=b*(e^n)^x, it is seen that e^n=m.

By taking the natural logarithm of each side of the equation, you get the following:

       LN(e^n)=LN(m).

From the definitions for natural logarithms: LN(e)=1 and LN(e)^n=n*LN(m), you can conclude that LN(e)^n=n.

By substituting n=LN(m) into LN(e^n)=LN(m), you get LN(e^n)=n.

Earlier it was stated that LN(e^n)=LN(m). By substituting LN(e^n)=n into this formula, you get the following result: n=LN(m).

If you want to use the exponential curve in the form y=b*e^(n*x), you should use the LOGEST() function with your data. The results of the LOGEST() evaluates the m values returned with the natural logarithm function LN().

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> Additional reference words: 2.00 3.00 4.00 4.00a

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Last reviewed: September 12, 1996
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