Given that log 6 ⁡ 2 = .387 ,find log 6 ⁡ 9I don't know proper method to solve this problem . Please help me.

Question

Given that       log    6    ⁡  2  =  .387 ,find       log    6    ⁡  9I don&#039;t know proper method to solve this problem . Please help me.

Anabelle Guzman · Accepted Answer

Let       log    6    ⁡  2  =  aHence,      log    6    ⁡  2  =      1          1      +              log        2            ⁡      3      and      log    6    ⁡  9  =            2              log        2            ⁡      3              1      +              log        2            ⁡      3        .Thus,      log    2    ⁡  3  =            1      −      a        a  and      log    6    ⁡  9  =                    2        (        1        −        a        )            a              1      +                        1          −          a                a              =  2  (  1  −  a  )  =  1.226...Now we see a best way:      log    6    ⁡  9  =      log    6    ⁡      36    4    =      log    6    ⁡  36  −      log    6    ⁡  4  =  2  −  2      log    6    ⁡  2  =  .  .  .

profesorluissp · Accepted Answer

This is probably the most brute-force approach, and it is reasoning like this (but with general variables rather than concrete numbers) that lies behind the formulas in the other answer.
We are after x such that

6^{x} = 9 2^{x} 3^{x} = 3^{2} 2^{x} = 3^{2 - x}

We are given that

6^{0.387} = 2 2^{0.387} 3^{0.387} = 2^{1} 2^{0.387 - 1} = 3^{- 0.387} 2^{0.613} = 3^{0.387}

Raising the last equation to the power of

x / 0.613

yields

2^{x} = 3^{0.387 x / 0.613}

but above we see that

2^{x}

at the same time is equal to

3^{x - 2}

. This means that we have

3^{x - 2} = 3^{0.387 x / 0.613}

, which can be solved relatively easily.

Given that log_6 2=.387 ,find log_6 9

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