I'm working on logarithmic equations and I'm stuck on how my book arrives at the next step.

Question

First, they use the change of base formula on,

log(sqrt(2))(x^3 - 2)
(sqrt(2)) is the base,changing to base 2

log(sqrt(2))(x^3 - 2)=
log2(x^3 - 2)/(log2(sqrt(2))

I understand that part.
The next step, the book has
log2(x^3 - 2)/(log2(sqrt(2))=
2 log2(x^3 - 2)

How did they get = 2 log2(x^3 - 2)??

I tried applying the different rules but I just can't get how they arrive at
= 2 log2(x^3 - 2)??

Please help. Thanks in advance.

Answer 1

look at the part log₂ √2

let x = log₂ √2
then 2^x = √2
2^x = 2^(1/2)
x = 1/2

so log2(x^3 - 2)/(log2(sqrt(2))
= log2(x^3 - 2)/(1/2)
= 2log2(x^3 - 2)

Answer 2

Thanks so much Reiny.

Answer 3

Now, I feel stupid.

I guess I should not work so late at night.

I tried all these rules when all I needed to do was simplify log2√2 using the basic definition/relationship !!

Thanks again. I don't remember learning logs some 40 years ago in HS and teaching myself.

Glad tutors like yourself take time to help, even with obvious (stupid) questions such as mine was.