# Logarithms: Change of Base

Sometimes a test or an assignment will tell you to solve something ridiculous like this:

17 = 5^{x}

And they want an actual approximate answer!

You might try to solve with this:

log_{5} 17 = *x*

But unless you’ve spent 150 bucks on a fancy calculator that solves for log_{5}, you won’t get an answer. But you can use a 12-dollar Wal-Mart calculator to solve this, if you know the change of base formula.

Remember that the “log” button on your simple calculator is actually log_{10}. So, using the change of base formula for

log_{5} 17 = *x*

You will get

Plugging that answer back into the original equation will give you

17 = 5^{1.760374428} = 17.00000001

which is a very good approximation.

When there is no subscript number after “log” you can assume that it means log_{10}.

Use the flashcards below to practice the change of base formula.

- log
_{2}3 =log 3 / log 2 - log
_{3}7 =log 7 / log 3 - log
_{7}11 =log 11 / log 7 - log
_{19}23 =log 23 / log 19 - log
_{29}31 =log 31 / log 29 - log
_{37}41 =log 41 / log 37