Question

How can I simplify `log_(sqrt(a))(a^2)` to get an exact numerical value?

Answer 1

`4`

Explanation:

We can start by applying the logarithm rule

`log_a(x^b)=blog_a(x)`

All this is saying is that we can bring the exponent out front. In our example, we get

`2log_sqrta(a)`

Next, we can rewrite `sqrta` as `a^(1/2)`. This helps us to leverage the following logarithm property:

`log_(a^b)(x)=1/b log_a(x)`

If we rewrite the expression as

`2log_(a^(1/2)) (a)`

Then our `color(blue)(b=1/2)` and `x=a`, and we now have

`2*1/(1/2)log_a(a)`

`=>4log_a(a)`

Log base `a` and `a` cancel, so we're just left with

`4`

Hope this helps1

Answer 2

`4`

Explanation:

we can just use the definition of logarithms

`log_ab=c=>a^c=b`

`"let "y=log_(sqrta)(a^2)`

using the defintion

`(sqrta)^y=a^2`

`=>a^((1/2)y)=a^2`

`:. y/2=2`

`=>y=2xx2=4`