Fill in each box below with an integer or a reduced fraction.

(a) [tex]\log _2 32=5[/tex] can be written in the form [tex]2^A=B[/tex] where
[tex]A= \square[/tex] and [tex]B= \square[/tex]

(b) [tex]\log _5 3125=5[/tex] can be written in the form [tex]5^C=D[/tex] where
[tex]C= \square[/tex] and [tex]D= \square[/tex]

Sure! Let's go through each part of the question step-by-step to find the values for the boxes.

(a) We have the logarithmic expression:

[tex]\[
\log_2 32 = 5
\][/tex]

We want to rewrite this in the exponential form [tex]\(2^A = B\)[/tex].

A logarithm statement [tex]\(\log_b a = c\)[/tex] can be interpreted as the base [tex]\(b\)[/tex] raised to the power of [tex]\(c\)[/tex] equals [tex]\(a\)[/tex] (i.e., [tex]\(b^c = a\)[/tex]).

Given [tex]\(\log_2 32 = 5\)[/tex], this means that:

[tex]\[
2^5 = 32
\][/tex]

So in the form [tex]\(2^A = B\)[/tex], we have:

- [tex]\(A = 5\)[/tex]
- [tex]\(B = 32\)[/tex]

(b) We have another logarithmic expression:

[tex]\[
\log_5 3125 = 5
\][/tex]

Again, we need to rewrite this in exponential form [tex]\(5^C = D\)[/tex].

Following the same principle as before, where [tex]\(\log_b a = c\)[/tex] means [tex]\(b^c = a\)[/tex],

Given [tex]\(\log_5 3125 = 5\)[/tex], this means that:

[tex]\[
5^5 = 3125
\][/tex]

So in the form [tex]\(5^C = D\)[/tex], we have:

- [tex]\(C = 5\)[/tex]
- [tex]\(D = 3125\)[/tex]

Therefore, the final answers are:

- For (a), [tex]\(A = 5\)[/tex] and [tex]\(B = 32\)[/tex]
- For (b), [tex]\(C = 5\)[/tex] and [tex]\(D = 3125\)[/tex]