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]