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
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
