1.8 KiB
1.8 KiB
https://www.youtube.com/watch?v=sULa9Lc4pck
log_a(b)
Pronounced log base a, this function is used to figure out what exponent you need to raise a to to get b.
log_ab = c can be rewritten as a^c = b.
5^{log_5^{(x+2)}}=x+2
By default, log refers to log_{10}. ln is shorthand for log_e.
\sqrt{x} = x^{1/2}
To get the reciprocal of a value, change the sign of the exponent.
x^{-1} = \frac{1}{x}
Domain
There are 3 places you need to worry about domain. The third is specific to logarithms.
- You can't divide by 0
- You can't take the square root of a negative without complex numbers
- You cannot take the
logof a zero, or a negative number.- There's no way to raise a number to an exponent and have it equal zero, or be a negative number.
- This can be used to help solve inequalities, because you know an equation that's wrapped in a logarithm must be $> 0$.
Finding the domain of added logarithms
log(x+2) + log(2x-3)
With the above example, you can find the domain of each function separately, then find the overlap of valid numbers.
Expanding logarithms
log_b(x*y) = log_b x + log_b y
log_b(\frac{x}{y}) = log_b x - log_b y
Example Problem:
log_5 z = 3
$$ log_5 y = 2 $$$$log_5(yz) = log_5 y + log_5 z 2 + 3 = 5 $$
Exponents can be moved to the front of a logarithm
log_3 x^5 = 5*log_3 x
Roots are just the inverse, so:
log_3 \sqrt x = \frac{1}{2}*log_3x
Change of base
log_b x = \frac{\log x}{\log b} = \frac{\ln x}{\ln b}
The above are all equivalent because the ratios are the same
The compound interest formula
A= Pe^{rt}
| Value | Description |
|---|---|
A |
Ending amount |
P or A_0 |
Starting amount |
r or k |
Rate (a %) |
t |
The amount of times interest is compounded |