Question 1

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

Question 2

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

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

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

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

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

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

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

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

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

Question 3

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

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