4.4 ­ Exponential and Logarithmic Equations.notebook
4.4: Exponential and Logarithmic Equations
November 10, 2016
Date: 11/10
An Exponential Equation is an equation containing a variable in an exponent
Examples:
*Some exponential equations can be solved by expressing each side of the equation as a power of the same base
*All exponential functions are one‐to‐one ‐ no two different ordered pairs have the same second component. So, if and 1
4.4 ­ Exponential and Logarithmic Equations.notebook
November 10, 2016
Ex 1: Solve each equation by expressing each side as a power of the same base:
a) b)
2
4.4 ­ Exponential and Logarithmic Equations.notebook
November 10, 2016
*Most exponential equations cannot be written so that each side has the same base.
Using Logarithms to Solve Exponential Equations:
Step 1: Isolate the exponential expression.
Step 2: Take the natural logarithm of both sides of the equation for bases other than 10. Take the common logarithm on both sides for base 10.
Step 3: Simplify using one of the following properties:
Step 4: Solve for the variable.
3
4.4 ­ Exponential and Logarithmic Equations.notebook
November 10, 2016
Ex 2: Solve:
a)
b)
4
4.4 ­ Exponential and Logarithmic Equations.notebook
November 10, 2016
Ex 3: Solve 5
4.4 ­ Exponential and Logarithmic Equations.notebook
November 10, 2016
Ex 4: Solve 6
4.4 ­ Exponential and Logarithmic Equations.notebook
Ex 5: Solve November 10, 2016
(Hint: This is a quadratic with
7
4.4 ­ Exponential and Logarithmic Equations.notebook
November 10, 2016
A logarithmic equation is an equation containing a variable in a logarithmic expression.
Examples:
Using the Definition of a Logarithm to Solve Logarithmic Equations:
Step 1: Express the equation in the form . (This requires a single log expression).
Step 2: Use the definition of logarithm to rewrite the equation in exponential form:
Step 3: Solve for the variable
*Step 4: Check the proposed solutions in the original equation. Include in the solution set only values for which ______
8
4.4 ­ Exponential and Logarithmic Equations.notebook
November 10, 2016
Ex 6: Solve:
a)
b)
9
4.4 ­ Exponential and Logarithmic Equations.notebook
November 10, 2016
Ex 7: Solve: 10
4.4 ­ Exponential and Logarithmic Equations.notebook
November 10, 2016
*Some logarithmic expressions can be expressed in the form __________________________ where the bases on both sides of the equation are the same. Because all logarithmic functions are one‐to‐one, we can conclude that __________.
Ex 8: Solve 11
4.4 ­ Exponential and Logarithmic Equations.notebook
November 10, 2016
Applications:
Ex 9: The risk of having a car accident increases exponentially as the concentration of alcohol in the blood increases. The risk is modeled by
, where x is the blood alcohol concentration and R, given by a percent, is the risk of having a car accident. What blood alcohol concentration corresponds to a 7% risk of a car accident? 12
4.4 ­ Exponential and Logarithmic Equations.notebook
November 10, 2016
Ex 10: Recall the compound interest formula
How long, to the nearest tenth of a year, will it take $1000 to grow to $3600 at 8% annual interest compounded quarterly?
13
4.4 ­ Exponential and Logarithmic Equations.notebook
November 10, 2016
Homework: pg. 457 #3 – 120 (multiples of 3)
14