Question

Select the correct answer.

Which exponential equation is equivalent to this logarithmic equation?

log^5 x - log^5 25 = 7

Answer 1

`\textsf{D.} \quad 5^9=x`

Explanation:

Given logarithmic equation:

`\log_5x-\log_525=7`

`\textsf{Apply the quotient log law}: \quad log_ax - \log_ay=\log_a \left((x)/(y)\right)`

`\implies \log_5\left((x)/(25)\right)=7`

`\textsf{Apply log law}: \quad \log_ab=c \iff a^c=b`

`\implies 5^7=(x)/(25)`

Multiply both sides by 25:

`\implies 25 \cdot 5^7=x`

Rewrite 5 as 5²:

`\implies 5^2 \cdot 5^7=x`

`\textsf{Apply exponent rule}: \quad a^b \cdot a^c=a^(b+c)`

`\implies 5^(2+7)=x`

`\implies 5^9=x`