Solving Exponential and Logarithmic Equations

ONE-TO-ONE PROPERTY OF EXPONENTIAL AND LOGARITHMIC EQUATIONS

Using these two rules can help solve equations that are considered to be "one-to-one". To classify an equation as one-to-one means that for every x value in an equation there is a specific y value designated for that one and only x value.

INVERSE PROPERTY OF EXPONENTIAL AND LOGARITHMIC EQUATIONS

If a>0 and a≠1:

• log_aa^x=x

• a^loga^x=x

Using these two rules can help you solve non one-to-one equations.

3.3 Properties of Logarithms

Change of Base Formula

Example:

Properties of Logarithms

- corresponds with the properties of its inverse: exponential functions

(These are non distributive functions, therefore, you cannot say something like: log(x + y) = log(x) + log(y)

Logarithms

~Logarithms are the inverses of exponentiation.

KEY TO EVERYTHING!

Example:

When the base numbers match, you can set the exponents equal to each other.

Log- The Common Log with a base of 10

Ln- The Natural Log with a base of e

Logarithm Graph

3.1 Exponential Functions and Their Graphs

Exponential Functions
Section 3.1 discusses exponential functions and their graphs.

Rules for Exponents

Polynomial and Exponential Functions Differ
In polynomial functions, a variable is raised to a term, while in an exponential functions, a term is raised to a variable.

Here a is the base and n is the exponent:


When two of the same bases are multiplied, the exponents are added together:


When two of the same bases are divided, the exponents are subtracted:


When a base and its exponent are raised by another power, the exponents are multiplied:


When the exponent is negative, the result if the reciprocal of the base and positive exponent:


Anything raised to the first power is equal to itself:

Anything raised to the zero power is one:

Exponential Functions and Their Graphs
The basic exponential function:

Here a is the base and x is the exponent. a is multiplied by itself x number of times.

Graphs of Exponential Functions:

The basic exponential function graph for the above exponential function (where a is the base and x is the exponent).

The y-intercept for all unmanipulated exponential functions is y=1.

The horizontal asymptote is at .
The domain is .
The range is .

Manipulating Exponential Functions:

describes the possible ways exponential functions and consequently their graphs can be manipulated.

When a is changed:

-The graph is vertically stretched by a factor of a if a>1.

-The graph is vertically compressed by a factor of a if a<1.

-If a is negative, the graph is reflected in the x-axis.

When b (the base) is changed:

-The y-intercept is unchanged

-The graph increases of decreases by a larger factor depending on whether b is less than or greater than 1, respectively.

When c exists:

-The graph moves right a factor of c when x-c.

-The graph moves left a factor of c when x+c.

When d exists:

-The graph moves up a factor of d when d is added.

-The graph moves down a factor of d when d is subtracted.

Examples:


Vertical compression.

Increased growth factor, shifted up 6 units (note change in y-intercept).

Compound Interest

When finding the compound interest, use:

A=Present value

P= Principal (initial value)

r=Annual interest rate (APR)

n= Number of compounding periods per year

t=Time in years

Example:

A total of $14,000 is invested at an annual interest rate of 8%. Find the balance after 4 years if it is compounded monthly:

Substitute 14,000 for P.

Substitute .08 for r (remember to convert 8% to decimal form).

Subsitute 12 for n (there are 12 months in 1 year).

Sustitute 4 for t.

Solve: $19,259.33

For continuously compounded interest, use:

e is the natural base, and is a button found on your calculator.

Example:

Using the same figures from the first example, the answer is: $19,279.79

6.2 Law of Cosines

You can solve an oblique triangle if you have all three sides (SSS) or two sides with their included angle (SAS) using the law of cosines.

Law of Cosines

Proof of Law of Cosines

http://pages.pacificcoast.net/~cazelais/173/law-sines-cosines.pdf

(page two)

The Law of Cosines can also be used to establish the following formula for the area of a triangle. This formula is called Heron's Area Formula.

Given any triangle with sides of lengths a, b, and c, the area of the triangle is:

Section 6.1 Law of Sines

The Law of Sines only find unknowns in oblique triangles. Oblique triangles are triangles that have no right angles.

The Law of Sines can be derived from an oblique triangle, by drawing an altitude from any side(h).

Using sine trigonometry you can break the triangle into two parts.

1.

2.

Now,it is possible to substitute for h. In order to have a trigonometry h needs to be eliminated.

Divide each side by c

Since you can draw an altitude from any of the sides all of the proportions are equal.

Example:

To find the area one can derive the equation from an oblique triangle with an altitude.

Since the altitude can be drawn from any side, the Area formula can change as long as it uses two sides and the included angle. Here are some other forms: