Why would anyone want to convert an exponential expression from one base to another? Why would logarithms help?

For that matter, what * are* exponents and logarithms?

If you simply want to learn the conversion formula, feel free to skip the introductions and link to the final page: **Finally: How to Convert the Base of an Exponent.**

## One Reason to Convert from one Exponential Base to Another

The chance of a tossed “fair” coin landing “heads” is 1/2. The chance of getting ten “heads” in ten tosses is “1/(2^10) = 1/1024”. That is not much worse than “1/1000 = 1/(10^3).”

Informally, that converted base-2 to base-10, and changed the exponent from ten to “approximately three.”

That’s easy enough if one selects a power of two that comes close to a power of ten, but surely there is a more accurate, general-purpose mathematical tool.

## Introduction to the Base and Exponent in Mathematics

Let’s answer that early question about “base”, “exponent” and “logarithm.”

Normally we use * base*-10 for everyday arithmetic. The ten digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} indicate specific values. Writing a number such as “twenty-seven and a half = 27.5” uses three of those digits. The value is found by using powers of ten.

“27.5 = 2*10 + 7*1 + 5/10 = 2*(10^1) + 7*(10^0) + 5*(10^-1)” is a very explicit way to show this value. The ‘*’ is the multiplication operator, and the ‘^’ is the * exponentiation* operator.

The * exponent* shows the number of times the

**base**is multiplied by itself. For any non-zero real number ‘b’ as the base and ‘j’ as a positive integer, “b^j = b*b*…*b” where the ‘b’ appears ‘j’ times. So “b^1 = b”.

By definition, “b^0 = 1” for any non-zero, real number ‘b’. Some would define “0^0 = 1” also, but others leave it undefined.

An exponent can be a negative integer. If ‘j’ is a positive integer, then “b^(-j) = 1/(b^j)”.

An exponent can be a fraction, or “rational” number. If ‘j’ and ‘k’ are positive integers, “b^(1/k) = the k-th root of b”. From the coin-toss example, “2 is the 10th root of 1,024.”

In other words, “b = ( b^(1/k) )^k”. Multiply the k-th root of ‘b’ by itself ‘k’ times, and you have the value ‘b’.

Then “b^(j/k) = the k-th root of b^j”.

Decoding Science. One article at a time.