Question

Question #fce3a

Answer 1

I am interpreting the question to mean that you want to find `log_9(sqrt(sqrt(sqrt(3))))`.

The answer to this is `1/16`.

Explanation:

We can expand `log_9(sqrt(sqrt(sqrt(3))))` to `log_9(((3^(1/2))^(1/2))^(1/2))` (since the square root is equivalent to taking the power of one half).

Now, `(a^b)^c=a^(bc)`. Using this, we simplify `log_9(((3^(1/2))^(1/2))^(1/2))` to `log_9(3^(1/2*1/2*1/2))=log_9(3^(1/8))`.

From here, we just need to remember the logarithm identity `log(a^b)=blog(a)`. Using this identity, we simplify `log_9(3^(1/8))` to `log_9(3)/8`.

But what is `log_9(3)`? Remember that taking the `log_b(a)` is finding a number `n` so that `b^n=a`. We know that `sqrt(9)=9^(1/2)=3`. So, `log_9(3)=1/2`.

Then, `log_9(3)/8=(1/2)/8=1/16`.