Download 3.4 – Exponential and Logarithmic Equations

Review and Warm Up
You may need to review the following properties as you try to work
through the Warm Up below.
One-to-One Properties
ax = ay if and only if x = y
loga x = loga y if and only if x = y
Inverse Properties
aloga x = x
loga ax = x
Warm Up - DO IN YOUR NOTES
Solve the following equations without a calculator.
a 2x = 32
b ln x − ln 3 = 0
x
c 13 = 9
d ex = 7
e ln x = −3
f log x = −1
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
1 / 19
3.4 – Exponential and Logarithmic Equations
Accelerated Pre-Calculus
Mr. Niedert
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
2 / 19
3.4 – Exponential and Logarithmic Equations
1
Solving Simple Equations
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
3 / 19
3.4 – Exponential and Logarithmic Equations
1
Solving Simple Equations
2
Solving Exponential Equations
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
3 / 19
3.4 – Exponential and Logarithmic Equations
1
Solving Simple Equations
2
Solving Exponential Equations
3
Solving Logarithmic Equations
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
3 / 19
Strategies for Solving Exponential and Logarithmic
Equations
Strategies for Solving Exponential and Logarithmic Equations
1
Rewrite the original equation in a form that allows the use of the
One-to-One Property of exponential or logarithmic functions.
2
Rewrite an exponential equation in logarithmic form and apply the
Inverse Property of logarithmic functions.
3
Rewrite an logarithmic equation in exponential form and apply the
Inverse Property of exponential functions.
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
4 / 19
Solving Exponential Equations
Example
Solve each equation and approximate the result to three decimal places if
necessary.
2
a e −x = e −3x−4
b 3 (2x ) = 42
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
5 / 19
Exact and Approximate Solutions
In (b) of the previous example we found an approximate solution of
about 3.807.
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
6 / 19
Exact and Approximate Solutions
In (b) of the previous example we found an approximate solution of
about 3.807.
If you were to find the exact solution then the exact solution is
log2 14.
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
6 / 19
Exact and Approximate Solutions
In (b) of the previous example we found an approximate solution of
about 3.807.
If you were to find the exact solution then the exact solution is
log2 14.
An exact solution is preferred when the solution is an intermediate
step in a larger problem. For a final answer, an approximate solution
is easier to comprehend.
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
6 / 19
Solving Exponential Equations
Practice
Solve each equation and approximate the result to three decimal places if
necessary.
2
a e −x = e 5x+6
b 4 (3x ) = 64
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
7 / 19
Solving Exponential Equations
Practice
Solve e x + 5 = 60 and approximate the result to three decimal places.
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
8 / 19
Solving Exponential Equations
Practice
Solve 2 32t−5 − 4 = 11 and approximate the result to the nearest
thousandth.
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
9 / 19
Solving and Exponential Equation of Quadratic Type
Example
Solve e 2x − 3e x + 2 = 0.
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
10 / 19
Solving and Exponential Equation of Quadratic Type
Practice
Solve e 2x − 7e x + 12 = 0.
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
11 / 19
Exponential and Logarithmic Equations (Part 1 of 2)
Assignment
Part 1: pg. 253-254 #1-4, 10-20 even, 26-62 EOE
Part 2: pg. 253-254 #5-8, 76-100 EOE
Assignment: pg. 253-254 #1-8, 10-20 even, 26-62 EOE, 76-100 EOE
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
12 / 19
Extraneous Solutions
When solving logarithmic equations, be sure to check your solution in
the original equation.
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
13 / 19
Extraneous Solutions
When solving logarithmic equations, be sure to check your solution in
the original equation.
We will need to make sure that the results do not yield extraneous
solutions.
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
13 / 19
Extraneous Solutions
When solving logarithmic equations, be sure to check your solution in
the original equation.
We will need to make sure that the results do not yield extraneous
solutions.
Remember that if you end up with extraneous solutions, they are not
considered to actually be solutions to the equation.
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
13 / 19
Solving Logarithmic Equations
Example
Solve each equation.
a ln x = 2
b log3 (5x − 1) = log3 (x + 7)
c log6 (3x + 14) − log6 5 = log6 2x
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
14 / 19
Solving Logarithmic Equations
Practice
Solve each equation.
a ln x =
2
3
b log4 (3x + 2) = log4 (6 − x)
c log3 (5x + 13) − log3 6 = log3 3x
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
15 / 19
Solving Logarithmic Equations
Example
Solve 5 + 2 ln x = 4 and approximate the result to three decimal places.
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
16 / 19
Solving Logarithmic Equations
Practice
Solve 2 log5 3x = 4.
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
17 / 19
Solving Logarithmic Equations
Practice
Solve log 5x + log(x − 1) = 2.
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
18 / 19
Exponential and Logarithmic Equations (Part 2 of 2)
Assignment
Part 1: pg. 253-254 #1-4, 10-20 even, 26-62 EOE
Part 2: pg. 253-254 #5-8, 76-100 EOE
Assignment: pg. 253-254 #1-8, 10-20 even, 26-62 EOE, 76-100 EOE
Accelerated Pre-Calculus
3.4 – Exponential and Logarithmic Equations
Mr. Niedert
19 / 19