Question

What is the true solution to 2 in e^in 5x = 2 in 15

Answer 1

We have to solve this equation:
$2 * ln e ^{ln(5x)}= 2 * ln 15/:2 \\ ln e ^{ln(5x)}=ln 15 \\ e ^{ln(5x)}=15$
5 x = 15
x = 15 : 5
Answer:
x = 3

Answer 2

Given equation $2\:ln\:e^{ln\left(5x\right)}\:=\:2\:ln\left(15\:\right).$

$\mathrm{Apply\:log\:rule}:\quad \:log_a\left(a^b\right)=b$

$\ln \left(e^{\ln \left(5x\right)}\right)=\ln \left(5x\right)$

$2\ln \left(5x\right)=2\ln \left(15\right)$

$\mathrm{Divide\:both\:sides\:by\:}2$

$\frac{2\ln \left(5x\right)}{2}=\frac{2\ln \left(15\right)}{2}$

$\ln \left(5x\right)=\ln \left(15\right)$

$\mathrm{For\:}\ln \left(5x\right)=\ln \left(15\right)\mathrm{,\:\quad solve\:}5x=15$

$5x=15$

$\mathrm{Divide\:both\:sides\:by\:}5$

$\mathrm{Verifying\:Solutions}:\quad x=3$

$\mathrm{Check\:the\:solutions\:by\:plugging\:them\:into\:}2\ln \left(5x\right)=2\ln \left(15\right)$

$\mathrm{Plug}\quad x=3:\quad 2\ln \left(5\cdot \:3\right)=2\ln \left(15\right)\quad \Rightarrow \quad \mathrm{True}$

$\mathrm{Therefore,\:the\:final\:solution\:for\:}2\ln \left(5x\right)=2\ln \left(15\right)\mathrm{\:is\:}$

$x=3$.

$x=3$