3.4 Solving Exponential and Logarithmic Equations

Introduction

There are two basic strategies you can use to solve these equations.

One-to-One Properties

 if and only if x=y

 if and only if x=y

Inverse properties

Examples of Exponential Equations:

Can be we written as:

By the one-to-one property, x=5

x=ln 7 by the Inverse property

Solving an Exponential Equation in Quadratic Form

   or   

          

          

                     x  =  0

Solving Logarithmic Equations

ln x = 2

Exponentiate each side

Final Answer Without a Calculator

Solving a Logarithmic Equation

Isolate the log (divide both sides by 2)

Exponentiate both side (by base 5)

Use the inverse property

Change-of-Base Formula

3.1 Exponential Functions and Their Graphs

Properties of Exponents:

When you multiply two terms, you add the exponents

When you divide two terms, you subtract the exponents

When you raise a power to a power, you multiply the exponents

When you raise a term to zero, it equals 1

Exponents to the power of n:

Definition of Exponential Function: 

Properties of an exponential function: 

Domain: 

Range: 

x-intercept: none

y-intercept: (0,1)

Horizontal Asymptote: 

Vertical Asymptote: none

Example:   

In the function above, the graph approaches the line y=0 and intersects the y-axis at (0,1).  

Shifting, Reflecting, and Stretching Exponential Functions

• a represents a vertical stretch or compress, if a is negative, the graph reflects in the x-axis
• b will make the graph's rate increase faster or slower
• c will shift the graph left or right
• d will shift the graph up or down

One-to-One Property

If the equation has the same base on both sides, the exponents can be set equal to each other and solved for x.

Example:

125 can be rewritten so it has the equation has a base of 5 on the right and left side.

Since both sides have a base of 5, the exponents can be set equal to each other to solve for x.