Mathematics: How do you solve this? Step by step please 2^x=5e^(x+1)

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How do you solve this? Step by step please 2^x=5e^(x+1)

Yoma321

02.02.2022

$x=\frac{\ln(5)+1}{\ln(2)-1}\approx-8.5039$

Step-by-step explanation:

So we have the equation:

$2^x=5e^{x+1}$

Let's take the natural log of both sides:

$\ln(2^x)=\ln(5e^{x+1})$

On the left, we can move the x to the front:

$x\ln(2)=\ln(5e^{x+1})$

On the right, we can separate the logs:

$x\ln(2)=\ln(5)+\ln(e^{x+1})$

For the second term on the right, move all the exponent stuff to the front:

$x\ln(2)=\ln(5)+(x+1)\ln(e)$

The natural log of e is 1. So:

$x\ln(2)=\ln(5)+(x+1)(1)$

Simplify:

$x\ln(2)=\ln(5)+x+1$

Subtract x from both sides:

$x\ln(2)-x=\ln(5)+1$

Factor out an x from the left:

$x(\ln(2)-1)=\ln(5)+1$

Divide both sides by the equation in the factor:

$x=\frac{\ln(5)+1}{\ln(2)-1}$

So, our answer is:

$x=\frac{\ln(5)+1}{\ln(2)-1}\approx-8.5039$

And we're done!

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