Exponents and Logarithms

Last updated on August 24, 2021 2 min read Year 11, Mathematics, Functions

Table of Contents

Exponentials

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 $e$ )
$e \approx 2.718281828459045235360 \dots$ (it’s irrational, like $π$ )

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, $A = (1 + \frac{1}{12})^{12} = 2.613 \dots$
- If your bank pays interest 10000 times a year, $A = (1 + \frac{1}{10000})^{10000} = 2.718 \dots$
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 $e = lim_{n \to \infty} (1 + \frac{1}{n})^{n}$ , which is very similar to the compound interest formula.
Therefore, $e$ is defined as the rate of continous compounding interest.

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:
- $e = \sum_{n = 0}^{\infty} \frac{1}{n!} = \frac{1}{1} + \frac{1}{1} + \frac{1}{1 \cdot 2} + \frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{1 \cdot 2 \cdot 3 \cdot 4} \dots$

$a^{b} = c \leftrightarrow l o g_{a} c = b$
Log is just another way of writing a power. It’s usually used to solve for $x$ when $x$ is in the power.
$\log_{e} a = \ln a$
$\log_{a} a = 1$
$l o g_{a} 1 = a$
$l o g_{a} b = \frac{l n (b)}{l n a} \leftarrow Change of Base Formula$
$l o g_{a} b + l o g_{a} c = l o g_{a} (b \cdot c)$
$l o g_{a} b - l o g_{a} c = l o g_{a} (\frac{b}{c})$
$l o g_{a} b^{c} = c \cdot \log_{a} b$
$l o g_{a} a^{x} = x$
10. $a^{l o g_{a} b} = b$
$\ln e^{x} = x$
$e^{l n x} = x$