Microsoft KB Archive/104449

= 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.