Let's solve the logarithmic equation step-by-step to find the equivalent exponential equation.
Given the logarithmic equation:
[tex]\[
\log_5 x - \log_5 25 = 7
\][/tex]
First, use the property of logarithms that allows you to combine the two logarithms on the left side:
[tex]\[
\log_5 \left(\frac{x}{25}\right) = 7
\][/tex]
Next, we need to convert the logarithmic equation to its exponential form. The general formula for converting a logarithmic equation [tex]\(\log_b (A) = C\)[/tex] to its exponential form is [tex]\(b^C = A\)[/tex].
In our case:
[tex]\[
\log_5 \left(\frac{x}{25}\right) = 7 \implies 5^7 = \frac{x}{25}
\][/tex]
Now, solve for [tex]\(x\)[/tex]:
[tex]\[
5^7 = \frac{x}{25}
\][/tex]
Multiply both sides of the equation by 25 to isolate [tex]\(x\)[/tex]:
[tex]\[
25 \cdot 5^7 = x
\][/tex]
Simplify the right hand side:
[tex]\[
25 \cdot 5^7 = 5^2 \cdot 5^7 = 5^{2+7} = 5^9
\][/tex]
Thus:
[tex]\[
5^9 = x
\][/tex]
Therefore, the correct exponential equation equivalent to the original logarithmic equation is:
[tex]\[
\boxed{5^9 = x}
\][/tex]
The correct answer is:
A. [tex]\(5^9 = x\)[/tex]
