Question

My teacher gave an example of converting a function into a logarithmic function and it is..
f(x)=106.16e^(-0.73x)
I understood the example up until
In y - In 106.16 = In e^(-0.73x)
But then the next step was
In y = In e^(-0.73x) + 4.66
I just have no clue where the 4.66 came from? He didn't explain the steps, just laid it out.

Answer 1

\[\large f(x)=106.16e^{(-0.73x)}\]Let f(x) = y\[\large y=106.16e^{(-0.73x)}\]Take the natural log of both sides.\[\large \ln y = \ln(106.16e^{(-0.73x)})\]

Answer 2

\[\large \ln y = \ln106.16+\ln(e^{(-0.73x)})\]

Answer 3

Looks like teacher simplified the first term on the right.
\[\large \ln 106.16=4.66494739...\]

Answer 4

\[\large \ln y = \ln(e^{(-0.73x)})+4.66\]

Answer 5

The logarithmic and exponential are INVERSE operations of one another. So since we are taking the natural log of an exponential e, the essentially "Cancel out".
\[\large \ln(e^{(-0.73x)})=(-0.73x)\]

Answer 6

\[\large \ln y = -0.73x + 4.66\]