SOLUTION: how do i evaluate the expression log[32](8) using the change of base formula

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Question 594731: how do i evaluate the expression log[32](8) using the change of base formula
Found 2 solutions by AnlytcPhil, jsmallt9:
Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!
log%2832%2C%288%29%29 = log%282%2C%288%29%29%2Flog%282%2C%2832%29%29 = log%282%2C%282%5E3%29%29%2Flog%282%2C%282%5E5%29%29 = 3%2F5 

That last step came from the principle log%28B%2C%28B%5EN%29%29=N. 

Edwin

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Logarithms can be evaluated "by hand" (without a calculator) if any of the following are true:
  1. The argument is a known power of the base.
  2. The base is a known power of the argument.
  3. The base and the argument are both known powers of some third number.

In the first case the answer is whatever that power is. For example log%284%2C+%2816%29%29+=+2 since 4%5E2+=+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.

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.

log%2832%2C+%288%29%29
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%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29, to change the base to 2:
log%2832%2C+%288%29%29+=+log%282%2C+%288%29%29%2Flog%282%2C+%2832%29%29
The two base 2 logs are case 1 logs. Since 2%5E5+=+32 and 2%5E3+=+8 these logs are 5 and 3, respectively. Now we have:
log%2832%2C+%288%29%29+=+log%282%2C+%2832%29%29%2Flog%282%2C+%288%29%29+=+3%2F5

FWIW: So what power of 32 is 8? Answer: 3/5