Use the Definition of a Logarithm to Solve Logarithmic Equations
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
-$$\ln \sqrt { x + 8 } = 7$$
A) $\left\{ \mathrm { e } ^ { 14 } - 8 \right\}$
B) $\left\{ \mathrm { e } ^ { 14 } + 8 \right\}$
C) $\left\{ \frac { \mathrm { e } ^ { 7 } } { 2 } + 8 \right\}$
D) $\left\{ e ^ { 7 } - 8 \right\}$

Question

Quizplus · Accepted Answer

To solve the equation $\ln \sqrt { x + 8 } = 7$, we first rewrite the square root in exponential form as $(x + 8)^{1/2}$, giving us $\ln (x + 8)^{1/2} = 7$. Using the property of logarithms that allows us to move the exponent to the front, we get $\frac{1}{2} \ln (x + 8) = 7$. Multiplying both sides by 2 gives $\ln (x + 8) = 14$. Converting from logarithmic to exponential form gives $x + 8 = e^{14}$, and solving for $x$ gives $x = e^{14} - 8$.

Use the Definition of a Logarithm to Solve Logarithmic Equations ln⁡x+8=7\ln \sqrt { x + 8 } = 7lnx+8​=7

Use the Definition of a Logarithm to Solve Logarithmic Equations $\ln \sqrt { x + 8 } = 7$