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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

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Solving Logarithmic Equations

2log 5x Original problem

2log 52 2x Exponentiate

52x Use inverse property to simplify …

32x Evaluate the exponential …

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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 …

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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

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Solving Logarithmic Equations