# Exponents and Logarithms

## Table of Contents

## Exponentials

- Exponentials are another type of function
- Exponentials have the general form $P=e^x$, where $a$ is a constant

### Euler’s Number $(e)$

In an exam, you’ll basically never get the easy version of exponentials

Instead you’ll get the Cool Kid version, which has this thing called Euler’s Number (also known as $e$)

$e\approx 2.718281828459045235360…$ (it’s irrational, like $\pi$)

To understand $e$, imagine a bank that pays 100% interest over 1 year.

In 1 year, $1 becomes $2.

Now, let’s say you want even better interest. You go to bank 2, which pays 50% interest, twice per year.

- At 6 months, you have $1.50.
- At 1 year, you have $2.25.
- Already you’re doing better!

You might remember the formula $A=(1+\frac{r}{n})^n$ from year 9 or 10. We can use this to find more profitable periods:

- If your bank gives monthly payments, $A=(1+\frac{1}{12})^{12}=2.613…$
- If your bank pays interest 10000 times a year, $A=(1+\frac{1}{10000})^{10000}=2.718…$

As you can see, even though the number of payments is increasing rapidly, the final amount is approaching the value of $e$.

This is because the formula for $e$ is $\displaystyle e=\lim_{n\rightarrow\infty} (1+\frac{1}{n})^n$, which is very similar to the compound interest formula.

Therefore, $e$ is defined as the rate of continous compounding interest.

### Properties of $e$

The gradient of the graph of $y=e^x$ is $e^x$ at every point on the graph

The area under the graph of $e^x$ is $e^x$ at every point on the graph

$e$ is the sum of the infinite series:

- $\displaystyle e=\sum_{n=0}^{\infty}\frac{1}{n!}=\frac{1}{1}+\frac{1}{1}+\frac{1}{1\cdot2}+\frac{1}{1\cdot2\cdot3}+\frac{1}{1\cdot2\cdot3\cdot4}…$

## Log Laws

$a^b =c \leftrightarrow log_a c=b$

Log is just another way of writing a power. It’s usually used to solve for $x$ when $x$ is in the power.

$\log_e a=\ln a$

$\log_a a=1$

$log_a 1=a$

$\color{lightblue}log_a b=\frac{ln(b)}{ln{a}}\color{lightgreen}\leftarrow\text{Change of Base Formula}$

$log_a b+log_a c=log_a(b\cdot c)$

$log_a b-log_a c=log_a(\frac{b}{c})$

$log_a b^c =c\cdot\log_a b$

$log_a a^x =x$

10.$a^{log_a b} =b$

$\ln e^x =x$

$e^{ln x}=x$