The Solution:
Given the logarithmic equation:
We are required to find the exponential equivalent of the given equation.
[tex]\begin{gathered} \log_5x-\log_525=7 \\ Applying\text{ the law: }\log_ax-\log_ay=\log_a(\frac{x}{y}) \end{gathered}[/tex]
We have:
[tex]\begin{gathered} \log_5(\frac{x}{25})=7 \\ \\ \frac{x}{25}=5^7 \\ \\ (25^{-1})^x=5^7 \end{gathered}[/tex]
By equalizing the bases on both sides, we get
[tex]\begin{gathered} 5^{-2x}=5^7 \\ \\ (5^{-2})^x=5^7 \\ \\ (\frac{1}{5^2})^x=5^7 \end{gathered}[/tex]
Cross multiplying, we get
[tex]\begin{gathered} x=5^2\times5^7=5^{2+7}=5^9 \\ \\ x=5^9 \end{gathered}[/tex]
Therefore, the correct answer is [option D]
