# How to Convert the Base of an Exponent with Logarithms

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## Who Were Euler and Napier?

Leonhard Euler: Portrait painted by Jakob Emanuel Handmann

Leonhard Euler was a Swiss mathematician who lived from 1707 to 1783.

One of his many contributions to mathematics was the proof that ‘e’ is an irrational number.

Euler’s formula for ‘e’ is “e^(i*x) = cosine(x) + i*sine(x)”, where ‘i’ is the square root of negative one, and ‘x’ is any real number.

Although ‘e’ may be named for the word “exponential” or in his honour as “Euler’s number”, he also has the completely different “γ (gamma) = Euler’s constant = 0.57721…”.

John Napier “invented” logarithms, and was particularly involved with natural logarithms.

Born in 1550, this Scottish landowner and polymath made other significant contributions to mathematics. He died in 1617, but several of his books were published posthumously.

By the way, he defined the “Naperian logarithm” which differs from the natural logarithm. His definition for ‘L’, the Naperian logarithm of a number ‘N’, is:

• “N = 10^7 * (1 – 10^(-7) )^L”

Enjoy solving for ‘L’.

## Finally, How to Convert the Base of an Exponent

Here is the general rule for converting an exponential expression from one base to another.

Let ‘b’ be the first base, and ‘a’ the target base. Let ‘m’ be the known exponent, and ‘x’ is the unknown.

To solve for ‘x’ when “b^m = a^x”, use

• x = m*ln(b)/ln(a)“.

If you plan to convert between the same bases, it’s worthwhile to calculate the conversion factor, “ln(b)/ln(a)”, and keep it for reference.

References:

Sondow, Jonathan and Weisstein, Eric W. e. (2012). MathWorld-A Wolfram Web Resource. Accessed October 7, 2012.

Weisstein, Eric W. Base; Logarithm; Natural LogarithmNapierian Logarithm(2012). MathWorld — A Wolfram Web Resource. Accessed October 7, 2012.

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