# Thread:

1. ## Logarithm question

If the sum is 2^x = 8, how do I work out x?

When I use my calculator, I press 'log', then 2 and then in brackets (8), so it looks like this "log2(8)". Why do I get the answer of 2.4082399.....

2. Originally Posted by yorkey
If the sum is 2^x = 8, how do I work out x?

When I use my calculator, I press 'log', then 2 and then in brackets (8), so it looks like this "log2(8)". Why do I get the answer of 2.4082399.....
There are a lot of different ways to solve this equation. I'm going to show you 2 of them:

1. Change 8 into a power of 2:

$2^x = 8~\implies~2^x=2^3$

Two powers of the same base are equal if the exponents are equal too.

2. Use logarithms (but correct) and the base-change-formula:

$2^x = 8~\implies~x=\log_2(8) ~\implies~x=\dfrac{\log(8)}{\log(2)} = \dfrac{\ln(8)}{\ln(2)}$

3. Using logs you can say

$\displaystyle 2^x = 8$

$\displaystyle \log_22^x = \log_28$

$\displaystyle x = \log_22^3$

$\displaystyle x = 3\log_22$

$\displaystyle x = 3\times 1$

$\displaystyle x = 3$

Seems like the long way home though, do you agree?

4. Easy peazy - now that I know what to do, and my questions won't get more complicated than this. Thank yuo!

5. By pressing log2(8) you're not saying base 2, index 8. Calculators by default use the 10 base. You'll have to use the base change formula, which is log of the index (by default base 10)/ log of the base (by default base 10).

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