College

If [tex]$5^a = 3125$[/tex], then the value of [tex]$5^{(a-3)}$[/tex] is:

(a) 25
(b) 125
(c) 625
(d) 1625

Answer :

Certainly! Let's go through this step-by-step.

First, we are given the equation:
[tex]\[ 5^a = 3125 \][/tex]

We need to find the value of [tex]\( a \)[/tex]. To do this, we can use the properties of logarithms. Specifically, we can take the logarithm (base 5) of both sides of the equation:
[tex]\[ \log_5 (5^a) = \log_5 (3125) \][/tex]

By using the logarithm property [tex]\(\log_b (b^x) = x\)[/tex], we have:
[tex]\[ a = \log_5 (3125) \][/tex]

Next, we need to calculate [tex]\( 5^{(a-3)} \)[/tex].

Given that [tex]\( a = 5.0 \)[/tex] (this is an exact value identified earlier), we can now substitute [tex]\( a \)[/tex] into the expression [tex]\( a - 3 \)[/tex]:
[tex]\[ a - 3 = 5 - 3 = 2 \][/tex]

Now we need to evaluate [tex]\( 5^2 \)[/tex]:
[tex]\[ 5^2 = 25 \][/tex]

Therefore, the value of [tex]\( 5^{(a-3)} \)[/tex] is:
[tex]\[ 25 \][/tex]

Hence, the correct answer is:
(a) 25