Question

which exponential equation is equivalent to this logarithmic equation?^log 5^x - ^log 5^25 =7picture of equation below

Answer 1

The Solution:

Given the logarithmic equation:

We are required to find the exponential equivalent of the given equation.

`\begin{gathered} \log_5x-\log_525=7 \\ Applying\text{ the law: }\log_ax-\log_ay=\log_a((x)/(y)) \end{gathered}`

We have:

`\begin{gathered} \log_5((x)/(25))=7 \\ \\ (x)/(25)=5^7 \\ \\ (25^(-1))^x=5^7 \end{gathered}`

By equalizing the bases on both sides, we get

`\begin{gathered} 5^(-2x)=5^7 \\ \\ (5^(-2))^x=5^7 \\ \\ ((1)/(5^2))^x=5^7 \end{gathered}`

Cross multiplying, we get

`\begin{gathered} x=5^2*5^7=5^(2+7)=5^9 \\ \\ x=5^9 \end{gathered}`

Therefore, the correct answer is [option D]