Properly using natural logs to solve for a variable

Question

I'm attempting to take a logistic graph and create an equation using the logistic model of continuous growth. I have taken the equation and simplified it down to

a • "e" = 3

I know that I need to pull out a natural log, but how do I do that if I am multiplying my e by an unknown variable (a)?

Answer 1

As you point out in your comment this is innane but:

$ae = 3 \iff $

$\ln ae = \ln 3 \iff$

$\ln a + \ln e = \ln a + 1 = \ln 3 \implies \ln a = \ln 3 - 1 = \ln (3/e)$ so

$a = e^{\ln a} = e^{\ln 3/e} = 3/e$.

Inane. But ... doable.

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Of course:

$ae = 3 \iff ae*(1/e) = 3*(1/e) \implies a = 3/e$

is a lot more straightforward and is much simpler.