Question

Simplify the following:

`log sqrt(32)/log 8`

Answer 1

Step 1: We know that

`(log ⁡a)/(log ⁡b) = log_b a`

So, `log sqrt(32)/log 8 = log_8 sqrt32`

Step 2: Express both numbers as powers of 2:

⇒ 32 = 2⁵

`sqrt(32) = (2^5)^(1/2)`

`sqrt(32) = 2^(5/2)`

⇒ 8 = 2³

Step 3: Substitute these powers into the logarithm:

`log_8 sqrt(32)`

= `log_(2^3) (2^(5/2))`

Step 4: Using the rule:

`log_(a^m) a^n = n/m`

We get,

`log_(2^3) 2^(5/2) = (5/2)/3`

`log_(2^3) 2^(5/2) = 5/6`