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.$a^{log_a b} =b$
$\ln e^x =x$
$e^{ln x}=x$
References
