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MissTica
3 years ago
10

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

Mathematics
2 answers:
Mkey [24]3 years ago
7 0
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
SVEN [57.7K]3 years ago
4 0

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


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