Question 594731
Logarithms can be evaluated "by hand" (without a calculator) if any of the following are true:<ol><li>The argument is a known power of the base.</li><li>The base is a known power of the argument.</li><li>The base and the argument are both known powers of some third number.</li></ol>
In the first case the answer is whatever that power is. For example {{{log(4, (16)) = 2}}} since {{{4^2 = 16}}}. For the other two cases, use the change of base formula. In case 2 change the base to whatever the argument is and in case 3 change the base to whatever the "third number" is.<br>
For other logarithms, you will need a calculator and you will use the change of base formula to change the base to base 10 or base e (ln) logarithms.<br>
{{{log(32, (8))}}}
If you are extremely clever you might be able to figure out what power of 32 is 8 (case 1). If not, then if you are somewhat clever you might be able to figure out what power of 8 is 32 (case 2). With a little effort you should be able to figure out that 32 and 8 are both powers of 2 (case 3). So we can evaluate this logarithm by using the change of base formula, {{{log(a, (p)) = log(b, (p))/log(b, (a))}}}, to change the base to 2:
{{{log(32, (8)) = log(2, (8))/log(2, (32))}}}
The two base 2 logs are case 1 logs. Since {{{2^5 = 32}}} and {{{2^3 = 8}}} these logs are 5 and 3, respectively. Now we have:
{{{log(32, (8)) = log(2, (32))/log(2, (8)) = 3/5}}}<br>
FWIW: So what power of 32 is 8? Answer: 3/5