Turning e into a Normal Expression

[Jason76]

I know how to get rid of natural log by taking e to the power of the natural log. However, how do you get rid of an e expression, so that you just have a normal expression (no e or \(\displaystyle \ln\))?

Say I had \(\displaystyle e^{4x}\) and I just wanted to convert to 4x.

 

[HallsofIvy]

I know how to get rid of natural log by taking e to the power of the natural log. However, how do you get rid of an e expression, so that you just have a normal expression (no e or \(\displaystyle \ln\))?

Say I had \(\displaystyle e^{4x}\) and I just wanted to convert to 4x.

What do you mean by "convert to 4x"? Obviously, \(\displaystyle e^{4x}\) is not equal to 4x so what connection do you want?

 

[Jason76]

What do you mean by "convert to 4x"? Obviously, \(\displaystyle e^{4x}\) is not equal to 4x so what connection do you want?

I mean \(\displaystyle e^{\ln4x}\) - e with the natural log as the exponent

First of all I got \(\displaystyle \ln 4x\). I want to change that to 4x.

\(\displaystyle e^{\ln 4x} = 4x\)

But if I had an expression of \(\displaystyle e^{4x}\), then how do I make that 4x?

I just looked up the answer: \(\displaystyle \ln(e^{4x}) = 4x\)

 

[daon2]

For instance \(\displaystyle e^{\ln4x} = 4x\) so how can we get 4x from \(\displaystyle e^{4x}\) ? At first we got 4x from the \(\displaystyle \ln\), but now trying to get it from the e.

\(\displaystyle e^x\) and \(\displaystyle \ln(x)\) are inverse functions, so the reverse composition is also true: \(\displaystyle \ln(e^{4x})=4x\). This can be easily verified using the properties of the logarithm.

 

[Jason76]

\(\displaystyle e^x\) and \(\displaystyle \ln(x)\) are inverse functions, so the reverse composition is also true: \(\displaystyle \ln(e^{4x})=4x\). This can be easily verified using the properties of the logarithm.

Also \(\displaystyle e^{\ln4x} = 4x\)