Exponential and Logarithmic Equations

There are two basic strategies for solving logarithmic equations. These strategies are based on two properties.

1. One-to-One Properties:

a^x=a^y if and only if x=y

log_ax=log_ay if and only if x=y

2.Inverse Properties

a^log_ax=x

log_aa^x=x

When solving both exponential and logarithmic equations, it is usually necessary to take either the log or natural log of both sides of the equation, (applicable to exponential equations) or exponentiate both sides of the equation. (applicable for logarithmic equations)

Exponential equation example:

(taking the natural log of both sides as seen in step 3)

Logarithmic equation example:

(exponentiation is seen in the step 2)

Its possible to solve exponential and logarithmic equations in quadratic forms and these are to be solved in the same method as other quadratic equations by factoring or the quadratic formula. It is important to check for extraneous solutions at the end of solving these types of problems.

3.2 Logarithms

3.2

Logarithms

Logarithmic functions are the inverses of exponential functions

This problem isn't too hard to solve:

5^x=125

5^x=5³

x=3

But what if it was:

5^x=124?

5 can't be raised to any whole number to make 124, so it isn't so easy to figure out. How can it be solved?

Logarithms can be used to solve this problem. Logarithms basically undo exponentiation.

In order to understand logarithms, there is one major property that needs to be known.

The Key To Everything

a^x=y if and only if log_ay=x

with numbers, this looks like:

10³=1000 if and only if log₁₀1000=3

The 10/a is called the BASE

The 100/y is the number that is found when raising the base to a power (the 3/x)

On a graphing calculator, there are two buttons off to the left side, one labeled LOG and the other LN

LOG= Common Log

· The common log means that the base of the equation is 10

LN= Natural Log

· The natural log means that the base of the equation is "e". e is an irrational number and it equals around 2.71828...

As logarithmic functions are inverses of exponential functions, their graphs are also inverses. This means that the logarithmic graph is an exponential graph reflected over the line y=x

A logarithmic function is a function in the form:

f(x)=a•log_b(x-c)+d

a: changes the y-coordinate, vertically stretches/compresses the graph, and if it is negative the graph is reflected in the x-axis

b: changes the base, changes the rate of growth and decay (if b is large, the graph grows slowly and if b is less than one, the graph changes to decay instead of growth)

c: shifts the graph left and right (counter intuitive)

d: shifts the graph up and down

3.3 Properties of Logarithms

Change of Base Formula

Base B

Base 10

How to use the change of base formula with common logarithms:

Base e

How to use the change of base formula with Natural Logarithms:

Properties of Logarithms

3.1 Exponential Functions and Their Graphs

n = exponent

x = base

Exponential Functions

Compound Interest

A= present value

P= initial value

r= interest rate (APR)

n= number of compounded periods per (ex. year, month, quarter)

t= time

Continuously compounded interest

Chapter 6.2 Law of Cosines

The law of cosines allows to solve for the values of the side lengths and angle measures. The law of cosines can be written in many ways, but the most common is:

also can be written as:

This just shows that you can use the law of cosine to solve for
all side lengths and can be written for all sides.

There is also an alternative writing where its solved in terms for the angle measures:

Also in this lesson we learned a very convenient method for solving the area of all kinds of Triangles.

Its called Heron's Area Formula :

but it's very important to remember what s is equal to.
s=(a+b+c)/2

Chapter 6.1 The Law of Sine's

The Law of Sine's is used for finding missing angles and side lengths on oblique triangles.

Deriving the Law of Sine's

Examples

Alternate Area Formula for Triangles

These three formulas are derived from the triangle above. They all equal the same thing and are used interchangeably.

1.

2.

3.