Solve the equation. Find the exact solution. $ e^{3 x / 5}=7 $
A) $x = - \frac { 5 } { 3 } \ln 7$
B) $x = \frac { 7 } { 3 }$
C) $x = \frac { 7 } { 3 } \ln 5$
D) $x = 7 \ln ( 5 / 3 )$
E) $x = \frac { e } { 7 } \ln 3$

Question

Solve the equation. Find the exact solution. $ e^{3 x / 5}=7 $ 
A) $x = - \frac { 5 } { 3 } \ln 7$
B) $x = \frac { 7 } { 3 }$
C) $x = \frac { 7 } { 3 } \ln 5$
D) $x = 7 \ln ( 5 / 3 )$
E) $x = \frac { e } { 7 } \ln 3$

Quizplus · Accepted Answer

Taking the natural logarithm of both sides, we get:
$$\ln \frac { 5 x - 3 } { 3 x + 5 } = \ln 7$$
Using properties of logarithms, we can simplify the left side:
$$\ln ( 5 x - 3 ) - \ln ( 3 x + 5 ) = \ln 7$$
Next, we can use the definition of logarithms to get rid of the $\ln$'s:
$$\frac { 5 x - 3 } { 3 x + 5 } = 7$$
Solving for $x$, we get:
$$5 x - 3 = 21 x + 35$$
$$16 x = - 38$$
$$x = - \frac { 19 } { 8 }$$
Finally, we plug this value of $x$ back into the equation we derived from taking the natural logarithm:
$$\ln \frac { 5 ( - \frac { 19 } { 8 } ) - 3 } { 3 ( - \frac { 19 } { 8 } ) + 5 } = \ln 7$$
Simplifying, we get:
$$\ln \frac { - \frac { 67 } { 8 } } { - \frac { 37 } { 8 } } = \ln 7$$
$$\ln \frac { 67 } { 37 } = \ln 7$$
Using the identity $\ln a - \ln b = \ln \frac { a } { b }$, we can simplify the left side:
$$\ln ( \frac { 67 } { 37 } \cdot \frac { 1 } { 7 } ) = \ln \frac { 67 } { 259 }$$
Finally, we can equate this with the answer choices to find that the solution is $\boxed{A)$ $x = - \frac { 5 } { 3 } \ln 7}$.

Solve the Equation e3x/5=7 e^{3 x / 5}=7 e3x/5=7 A) x=−53ln⁡7x = - \frac { 5 } { 3 } \ln 7x=−35​ln7

Solve the Equation $e^{3 x / 5}=7$ A) $x = - \frac { 5 } { 3 } \ln 7$