# Getting to grips with simple logarithmic equations

Please, help me to understand my mistakes, and the logic, so I can once and for all understand and remember nuances - they do seem to slip away as the time passes; I thought this topic is clear to me, but, well, I see it's not. There are two questions.

(1) Here is the example I found on the web:

$\$100$go to$\$150$ in $5$ years

To see how, they suggest to:

$\dfrac{\ln\left(\frac{150}{100}\right)}{5} = 8.1\%$

When I try to see how they got here and why their approach "works", I get stuck - I don't see how they came to the natural logarithm (base epsilon) and how their math came about. I assume they started from something like this:

$100 \cdot Y^5 = 150$ (although usually it would be $100 \cdot (1 + x\%)^5 = 150$)

$Y^5 = \dfrac{150}{100}$

$\log_Y 1.5 = 5$ (here I assume it can be $\ln$ or $\log$ - it shouldn't matter as far as I know (which is very little) about both)

I don't see how it should evolve.

Then I decided to use simple math to see if I get the same $\%$ without using logarithm.

$100 \cdot (1 + x)^5 = 150$

$(1 + x)^5 = 1.5$

$x = 1.5^{1/5} - 1 = 8.4\%$

which is more or less close to $8.1\%$

(2) I decided to use another path, and decided to use their result $8.1\%$ to see if I get $5$ years to get from $100$ to $150$ with the given percent. So I did the following:

$100 \cdot (1 + 0.081)^N = 150$

Then, using logarithm base $10$, I indeed get almost $5$ years:

$(1 + 0.081)^N = 1.5$

$\log_{1.081} 1.5 = 5.2$