SOLUTION: Log0.25 base 8

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Question 1175406: Log0.25 base 8
Found 3 solutions by Theo, MathTherapy, greenestamps:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
log of .25 to the base of 8 is found as follows:

use the base conversion formula to convert the log to base 10.

the conversion formula says that log .25 to the base of 8 is equal to log .25 to the base of 10 divided by log(8) to the base of 10.

therefore:

log8(.25) = log(.25)/log(8) using the log function of your calculator because the log function of your calculator is set to base 10, i.e. log(.25) in your calculator is really log10(.25).

using the calculator, log(.25) to the base of 8 is equal to -2/3.

the log base conversion formula can also be derived in the following manner.

start with log8(.25) = y

this is true if and only if 8^y = .25

take the log of both sides of this equation to get:

log(8^y) = log(.25)

by properties of logarithms, this becomes:

y * log(8) = log(.25)

solve for y to get:

y = log(.25) / log(8).

your answer was log8(.25) = -2/3.

by basic definition of logs, this is true if and only if 8^(-2/3) = .25.

you can use your calculator to verify that 8^(-2/3) = .25.

note that, by laws of exponents, 8^(-2/3) = 1/8^(2/3).

you can use your calculator to verify that 1/8^(2/3) = .25.

all has been confirmed by me through the use of my ti-84 plus calculator.

your solution is that log8(.25) = -2/3.


Answer by MathTherapy(10895) About Me  (Show Source):
You can put this solution on YOUR website!
Log0.25 base 8
************
Let log+%288%2C+%280.25%29%29+=+m
           Then: 8%5Em+=+.25
               %282%5E3%29%5Em+=+25%2F100
                2%5E%283m%29+=+1%2F4 ---- Applying %28b%5Ee%29%5Er+=+b%5E%28er%29, and reducing 25%2F100 to 1%2F4
                2%5E%283m%29+=+2%5E%28-+2%29 ---- Converting 1%2F4 to 2%5E%28-+2%29
                    3m = - 2 -- EXPONENTS are equal, since bases are equal
                      m = highlight%28-+2%2F3+=+log+%288%2C+%280.25%29%29%29

Answer by greenestamps(13391) About Me  (Show Source):
You can put this solution on YOUR website!


log%288%2C%28.25%29%29

We can evaluate this using the change-of-base formula for logarithms:

log%28a%2C%28x%29%29=log%28b%2C%28x%29%29%2Flog%28b%2C%28a%29%29

With the change-of-base formula, the new base "b" can be any base we want. Since in the given expression "8" and ".25" are both powers of 2, we can use b=2 for the base.

log%288%2C%28.25%29%29=log%282%2C%28.25%29%29%2Flog%282%2C%288%29%29=%28-2%29%2F%283%29

ANSWER: -2/3