How to Convert the Base of an Exponent with Logarithms


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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

Converting base-2 to base-1o for a coin toss : Image by ICMA Photos

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”.

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