AP Precalculus 2.8-2.11 (logarithms)

Logarithm function form

y = a * log (x-h) + k

How to find the VA in a logarithmic function

Set whatever is inside the parentheses greater than 0, and solve for x

If 'b' is greater than one...

the function is increasing, concave down

If 'b' is between 0 and 1...

the function is decreasing, concave up

Product property

log (M*N) = log M + log N

Quotient property

log (M/N) = log M - log N

Power property

log N^p = p*log N

How to solve exponential equations

1. Isolate exponential to one side

2. Make like bases

3. Drop bases and set exponents equal to each other

4. Solve equation

How to solve logarithmic equations

1. Must have same base

2. Drop logs and solve equation

3. Check for extraneous solutions with OG EQUATION!!!

Properties of exponential inequalities

- If base is greater than one, solve like normal

- If base is between 0 and 1, flip inequality sign when you drop the bases

Properties of logarithmic inequalities

- If base is greater than one, than log x is greater than log y if x is greater than y

- If base is between 0 and 1, and x is less than y, then log x is less than log y

- If base is between 0 and one is y is greater than x, then log x is greater than log y

Consider the ________ of logs before you answer

domain!!!

What if it has a log on only one side?

When changing to exponential, the answer must STAY ON THE SAME SIDE OF THE INEQUALITY THE WHOLE TIME

Richter scale model

R(x) = log (x/x base 0)

- measures magnitude of earthquake

- x base 0 = 10^-3

pH scale model

pH = -log [H^+]

[H^+] = 10^-pH

-measures how acidic or basic a substance ranks

- [H^+] is the concentration of hydrogen ions (the plus just makes it fancy, don't worry about it just write it in the answer)

Newton's Law of Cooling

T(t) = T (base m) + (T base 0 - T base m) e^-kt

- the temperature (T) of an object at time (t)

- T base m = temp of surrounding medium

- T base 0 = initial temp of object

- k = constant (often what you're solving for)

Unit 2 part 2 themes and verses

- logarithmic functions

- becoming Christ-like (increasing holiness)

- verses: Romans 12:2 & Ephesians 4:22-24