Download 171S5.5o Solving Exponential and Logarithmic Equations

171S5.5o Solving Exponential and Logarithmic Equations
MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 5: Exponential and Logarithmic Functions
5.1 Inverse Functions
5.2 Exponential Functions and Graphs
5.3 Logarithmic Functions and Graphs
5.4 Properties of Logarithmic Functions
5.5 Solving Exponential and Logarithmic Equations 5.6 Applications and Models: Growth and Decay; and Compound Interest
Solving Exponential 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 Property
For any a > 0, a ≠ 1, November 21, 2011
5.5 Solving Exponential and
Logarithmic Equations
• Solve exponential equations.
• Solve logarithmic equations.
Example
Solve
Solution:
Write each side as a power of the same number (base).
Since the bases are the same number, 2, we can use the base­exponent property and set the exponents equal:
Check x = 4:
ax = ay if and only if x = y.
The solution is 4.
TRUE
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171S5.5o Solving Exponential and Logarithmic Equations
Another Property
November 21, 2011
Example
Solve: 3x = 20.
Property of Logarithmic Equality
For any M > 0, N > 0, a > 0, and a ≠ 1,
loga M = loga N This is an exact answer. We cannot simplify further, but we can approximate using a calculator.
M = N.
We can check by finding 32.7268 ≈ 20.
Solving Logarithmic Equations
Example
Solve: e0.08t = 2500.
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.
The solution is about 97.8.
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171S5.5o Solving Exponential and Logarithmic Equations
Example
November 21, 2011
Example
Solve: log3 x = −2.
Solve:
Check:
Solution:
TRUE
The solution is
Example
Example (continued)
Check x = 2:
Check x = –5:
Solve:
Solution:
FALSE
TRUE
The number –5 is not a solution because negative numbers do not have real number logarithms. The solution is 2.
Only the value 2 checks, and it is the only solution.
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171S5.5o Solving Exponential and Logarithmic Equations
Example ­ Using the Graphing Calculator
November 21, 2011
Suggestions for solving exponential and logarithmic equations:
Solve: e0.5x – 7.3 = 2.08x + 6.2.
Exponential equation: Write so that bases are equal, set exponents equal, and solve, if possible. If not possible, write as logarithmic equation and solve.
Solve:
Graph y1 = e0.5x – 7.3 and y2 = 2.08x + 6.2 and use the Intersect method.
Base­Exponent Property
For any a > 0, a ≠ 1, ax = ay if and only if x = y.
The approximate solutions are –6.471 and 6.610.
Solve the exponential equation algebraically. Then check using a graphing calculator.
444/4. 37x = 27
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,
loga M = loga N if and only if M = N.
loga x = y if and only if x = ay
Solve the exponential equation algebraically. Then check using a graphing calculator.
444/10. 4
171S5.5o Solving Exponential and Logarithmic Equations
November 21, 2011
Solve the exponential equation algebraically. Then check using a graphing calculator.
444/20. 1000e0.09t = 5000
Solve the exponential equation algebraically. Then check using a graphing calculator.
444/28. 2x+1 = 52x
Solve the logarithmic equation algebraically. Then check using a graphing calculator.
444/36. log5 (8 ­ 7x) = 3
Solve the logarithmic equation algebraically. Then check using a graphing calculator.
444/48. log5 (x + 4) + log5 (x ­ 4) = 2
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171S5.5o Solving Exponential and Logarithmic Equations
Use a graphing calculator to find the approximate solutions of the equation.
444/54. 0.082e0.05x = 0.034
November 21, 2011
Use a graphing calculator to find the approximate solutions of the equation.
444/62. log5 x + 7 = 4 ­ log5 x
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