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

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.

  • Logarithms -

    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)

  • Logarithms -

    Thanks so much Reiny.

  • Logarithms -

    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.

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