Solve the exponential equation. Express the solution set in terms of natural logarithms.
-$$4 ^ { x + 4 } = 5 ^ { 2 x + 5 }$$
A) $\{ 7 \ln 5 - 5 \ln 4 \}$
B) $\{ \ln 5 - \ln 4 \}$
C) $\left\{ \frac { 5 \ln 5 - 4 \ln 4 } { \ln 4 - 2 \ln 5 } \right\}$
D) $\left\{ \ln \left[ \frac { 5 ^ { 5 } } { 4 ^ { 4 } } - \frac { 4 } { 5 ^ { 2 } } \right] \right\}$

Question

Quizplus · Accepted Answer

To solve the given exponential equation $4^{x+4} = 5^{2x+5}$, we first take the natural logarithm of both sides to get $\ln(4^{x+4}) = \ln(5^{2x+5})$. Using the logarithmic property $\ln(a^b) = b\ln(a)$, we can rewrite this as $(x+4)\ln(4) = (2x+5)\ln(5)$. Rearranging the terms to solve for $x$, we get $x\ln(4) - 2x\ln(5) = 5\ln(5) - 4\ln(4)$, which simplifies to $x(\ln(4) - 2\ln(5)) = 5\ln(5) - 4\ln(4)$. Solving for $x$ gives $x = \frac{5\ln(5) - 4\ln(4)}{\ln(4) - 2\ln(5)}$, which matches choice C.

Solve the Exponential Equation 4x+4=52x+54 ^ { x + 4 } = 5 ^ { 2 x + 5 }4x+4=52x+5

Solve the Exponential Equation $4 ^ { x + 4 } = 5 ^ { 2 x + 5 }$