Sections 3.3, 14.3

Logarithms

Inverse Logarithms

The last thing you need to be able to do is take inverse logarithms. This is necessary for problems where you are presented with the problem where the unknown is in the logarithm. For example (again using base-10 logs):

Problem: logx = 2

This problem is asking: the logarithm of what number is 2? We have learned before that the logarithm of a number is the exponent you would place on 10 to get that number. So the simplest way to reword this question is: If you take 2 and use it as the exponent on 10, what number will that give you:

x = 10² = 100

In algebraic terms, what you are doing to solve this problem is getting rid of the logarithm taking both sides of the equation and placing them as exponents on the number 10:

logx = 2

10^logx = 10²

Since a logarithm is the opposite of raising something to the power of 10, these cancel each other out:

~~10^log~~^x = 10²

x = 10²

x = 100

Calculating inverse logarithms on a calculator proceeds much like before: enter the number you want to take the inverse logarithm of, then press the key combination for a 10^x function. This usually is done by pressing the "INV" key (inverse) or "2^cd" key, then pressing the button that has "log" on it, with " 10^x" written above it. For example, find the value of x:

Problem: logx = 1.68

Enter 1.68 in your calculator.

Now press the "INV or "2cd" button, then the " 10^x" button:

x = 10^1.68

x = 47.9

Again, ask yourself if this number makes sense. You know that the logarithm of 100 is 2, and the logarithm of 10 is 1, so your result that the logarithm of 1.68 is 47.9 seems reasonable.