table of contents solving logarithmic equations a logarithmic equation is an equation with an...
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Table of Contents
Solving Logarithmic Equations
• A logarithmic equation is an equation with an expression that contains the log of a variable expression. The following examples will show how to solve this type of equation.
• Example 1:
Solve the equation .5xlog2
Exponentiate both sides of the equation, using base 2. In other words, write both side of the equation as exponents of base 2.
• Method A
Table of Contents
Solving Logarithmic Equations
2log 5x Original problem
2log 52 2x Exponentiate
52x Use inverse property to simplify …
32x Evaluate the exponential …
Table of Contents
Solving Logarithmic Equations
Use the definition of a log to express the equation in exponential form.
52x
• Method B
2log 5x Original problem
32x Evaluate the exponential …
Table of Contents
Solving Logarithmic Equations
• Example 2:Solve the following equation. 217)3x2ln(5
Get the log expression on the left side by itself.
14)3x2ln(5
514)3x2ln(
Note that ln is a log with base e.
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Solving Logarithmic Equations
Method A
145
145
145
14ln(2 3) 514log (2 3) 5
2 3
2 3
3
2
e
x
x
x e
x e
ex
14ln(2 3) 5
145
145
145
14ln(2 3) 5
2 3
2 3
3
2
x
x
e e
x e
x e
ex
Method B
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Solving Logarithmic Equations
• Example 3:Solve the following equation (nearest hundredth).
1)4xlog()2xlog(
1)4x)(2x(log Use the Product property to combine the logs.
Note that “log” is a log with base 10. Now either exponentiate (Method A) or use the definition of a log (Method B) to get …
110)4x)(2x(
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Solving Logarithmic Equations
108x2x2 Use the quadratic formula to solve for x.
018x2x2
1912
1922
2
762x
Written as decimals (nearest hundredth), the solutions are ...
5.36, 3.36x
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Solving Logarithmic Equations
5.36, 3.36x 1)4xlog()2xlog(
• Go back and look at the original equation and compare with the answers.
• Note that for the solution x = - 5.36, the expressions (x - 2) and (x + 4) will both be negative. Since the logarithm of a negative number is undefined, we must throw this extraneous solution out.
• The solution to the equation is 3.36x
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Solving Logarithmic Equations
• Example 4:Solve the following equation.
8ln)4xln(xln
8ln4x
xln
Use the Quotient property to combine the logs on the left hand side of the equation.
Equating the two expressions yields ... 84x
x
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Solving Logarithmic Equations
32x8)4x(8x
Solve for x.
7
32x
Table of Contents
Solving Logarithmic Equations