notes/education/math/MATH1050/Exponents/Logarithms.md

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 log of 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