SOLUTION: How do I evaluate this expression using the properties of logarithms? 2log₈4 – 1/3 log₈8

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: How do I evaluate this expression using the properties of logarithms? 2log₈4 – 1/3 log₈8      Log On


   

Question 404283: How do I evaluate this expression using the properties of logarithms?
2log₈4 – 1/3 log₈8
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
2%2Alog%288%2C4%29+-+%281%2F3%29%2Alog%288%2C8%29
Use the general rule:
a%2Alog%28b%29+=+log%28b%5Ea%29
log%288%2C4%5E2%29+-+log%288%2C8%5E%281%2F2%29%29
Now use the rule:
log%28a%29+-+log%28b%29+=+log%28a%2Fb%29 (logs with the same base)
log%288%2C+%2816%2Fsqrt%288%29%29%29
log%288%2C+%288%2Fsqrt%282%29%29%29
Now you can use the reverse of the above rule:
log%28a%2Fb%29+=+log%28a%29+-+log%28b%29
log%288%2C8%29+-+log%288%2Csqrt%282%29%29
Rule:
log%28a%2Ca%29+=+1
1+-+log%288%2Csqrt%282%29%29
Now, can I get sqrt%282%29 in terms of 8?
I know 8%5E%281%2F3%29+=+2
so, sqrt%288%5E%281%2F3%29%29+=+sqrt%282%29
This is the same as:
8%5E%281%2F6%29+=+sqrt%282%29
therefore
1+-+log%288%2Csqrt%282%29%29+=+1+-+log%288%2C8%5E%281%2F6%29%29
rule:
log%28a%2Ca%5Eb%29+=+b
1+-+1%2F6+=+5%2F6 answer
Unless I goofed