171S5.5q Solving Exponential and Logarithmic Equations

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April 18, 2013

Nov 17­11:31 AM

CHAPTER 5: Exponential and Logarithmic Functions

5.1 Inverse Functions5.2 Exponential Functions and Graphs5.3 Logarithmic Functions and Graphs5.4 Properties of Logarithmic Functions5.5 Solving Exponential and Logarithmic Equations 5.6 Applications and Models: Growth and Decay; and Compound Interest

MAT 171 Precalculus AlgebraDr. Claude Moore

Cape Fear Community College

Solving Exponential Equations contains an example of solving when bases can be made equal.

Solving Exponential Equations contains an example of solving when bases can NOT be made equal.

Three animations in CourseCompass are available.

Solving Logarithmic Equations contains an example of solving by taking the logarithm of both sides of the equation.

Solving Logarithmic Equations is a very good (3­minute) YouTube video.http://www.youtube.com/watch?v=59j0ALU3N7k&feature=relmfu

Solving Exponential Equations is a very good (11­minute) YouTube video.http://www.youtube.com/watch?v=7BJ2MlRMZ14&feature=related

YouTube Videos appropriate for Section 5.5:

Nov 17­11:31 AM

5.5 Solving Exponential andLogarithmic Equations

• Solve exponential equations.• Solve logarithmic equations.

Equations with variables in the exponents, such as 3x = 20 and 25x = 64,are called exponential equations.

Use the following property to solve exponential equations.

Base­Exponent PropertyFor any a > 0, a ≠ 1,

ax = ay if and only if x = y.

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Example

Solution:Write each side as a power of the same number (base).

Solve

Since the bases are the same number, 2, we can use the base­exponent property and set the exponents equal:

Check x = 4:

TRUEThe solution is 4.

Another Property

Property of Logarithmic Equality

For any M > 0, N > 0, a > 0, and a ≠ 1,

loga M = loga N M = N.

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Example

Solve: 3x = 20.

This is an exact answer. We cannot simplify further, but we can approximate using a calculator.

We can check by finding 32.7268 ≈ 20.

Example

Solve: e0.08 t = 2500.

The solution is about 97.8.

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Solving Logarithmic Equations

Equations containing variables in logarithmic expressions, such as log2 x = 4 and log x + log (x + 3) = 1,

are called logarithmic equations.

To solve logarithmic equations algebraically, we first try to obtain a single logarithmic expression on one side and then write an equivalent exponential equation.

Example

Solve: log3 x = −2.

TRUE

Check:

The solution is

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ExampleSolve:

Solution:

Check x = –5:

FALSE

Check x = 2:

TRUE

The number –5 is not a solution because negative numbers do not have real number logarithms. The solution is 2.

171S5.5q Solving Exponential and Logarithmic Equations

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April 18, 2013

Nov 17­11:31 AM

ExampleSolve:

Solution:

Only the value 2 checks, and it is the only solution.

Example ­ Using the Graphing Calculator

Solve: e0.5x – 7.3 = 2.08x + 6.2.

Solve:Graph y1 = e0.5x – 7.3 and y2 = 2.08x + 6.2 and use the Intersect method.

The approximate solutions are –6.471 and 6.610.

Nov 18­6:48 AM

Suggestions for solving exponential and logarithmic equations:

Exponential equation: Write so that bases are equal, set exponents equal, and solve, if possible. If not possible, write as logarithmic equation and solve.Base­Exponent PropertyFor any a > 0, a ≠ 1, a x = a y if and only if x = y.

Logarithmic equation:If bases of logs are equal, set quantities equal and solve. If not possible, write as exponential equation and solve.Property of Logarithmic Equality: For any M > 0, N > 0, a > 0, and a ≠ 1,

log a M = log a N if and only if M = N .

log a x = y if and only if x = a y

Nov 17­12:40 PM

Solve the exponential equation algebraically. Then check using a graphing calculator.452/2. 2 x = 32

Solve the exponential equation algebraically. Then check using a graphing calculator.452/4. 3 7x = 27

Nov 17­12:40 PM

Solve the exponential equation algebraically. Then check using a graphing calculator.452/6. 3 7x = 27

Solve the exponential equation algebraically. Then check using a graphing calculator.452/10.

Nov 17­12:40 PM

Solve the exponential equation algebraically. Then check using a graphing calculator.452/14. 15 x = 30

Solve the exponential equation algebraically. Then check using a graphing calculator.452/20. 1000e 0.09t = 5000

Nov 17­12:40 PM

Solve the exponential equation algebraically. Then check using a graphing calculator.452/24. 250 ­ (187) x = 0

Solve the exponential equation algebraically. Then check using a graphing calculator.452/28. 2 x+1 = 5 2x

171S5.5q Solving Exponential and Logarithmic Equations

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April 18, 2013Solve the logarithmic equation algebraically. Then check using a graphing calculator.452/40. log 5 (8 ­ 7x) = 3

Solve the logarithmic equation algebraically. Then check using a graphing calculator.452/44. log ( x + 5) ­ log ( x ­ 3) = log 2

Solve the logarithmic equation algebraically. Then check using a graphing calculator.452/52. ln ( x + 1) ­ ln x = ln 4

Solve the logarithmic equation algebraically. Then check using a graphing calculator.452/54. log 5 (x + 4) + log 5 (x ­ 4) = 2

Solve: 452/58.

Solve: 452/60. 2 ln x ­ ln 5 = ln ( x + 10)

Solve: 452/61. e x ­ 2 = ­ e ­x

Solve: 452/62. 2 log 50 = 3 log 25 + log ( x ­ 2)

Use a graphing calculator to find the approximate solutions of the equation.452/64. 0.082e 0.05x = 0.034

Use a graphing calculator to find the approximate solutions of the equation.452/66. 4x ­ 3 x = ­6

Use a graphing calculator to find the approximate solutions of the equation.452/68. 4 ln ( x + 3.4) = 2.5

Use a graphing calculator to find the approximate solutions of the equation.452/72. log 3 x + 7 = 4 ­ log 5 x