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7-4 Study Guide and InterventionSolving Logarithmic Equations and Inequalities

Solving Logarithmic Equations

Property of Equality for 

Logarithmic Functions

If b is a positive number other than 1,

then logb  x = logb  y if and on! if x = y "

Example 1: Solve log2  2 x = 3.

log2  2 x = 3 Ori#ina e$uation

2 x = 23

Definition of o#arithm

2 x = 8 %impif!"

 x = 4 %impif!"

The solution is x = 4.

Example 2: Solve the equationlog2  ( x + 17 = log2  (3 x + 23.

Since the bases of the logarithms are equal, ( x + 17

must equal (3 x + 23.

( x + 17 = (3 x + 23

 !" = 2 x

 x = !3

&hapter ' 26 Glencoe Algebra

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Exercises

Solve each equation.

1. log2  32 = 3 x 2. log3  2c = !2

3. log2 x  1" = !2 !. log25   ( x2 )  =

12

". log4  (# x + 1 = 2 #. log8  ( x ! # =2

3

7. log4  (3 x ! 1 = log4  (2 x + 3 $. log2  (   x2  ! " = log2  (2 x + 2

%. log x   +log4  27 = 3 1&. log2  ( x + 3 = 4

11. log x  1$$$ = 3 12. log8  (4 x + 4 = 2

13. log2  x = log2  12 1!. log3  ( x ! # = log3  13

1". log10   x = log10  (# x ! 2$ 1#. log5   x = log5  (2 x ! 1

17. log4  ( x + 12 = log4  4 x 1$. log6  ( x ! 3 = log6  2 x

&hapter ' 27 Glencoe Algebra

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7-4 Study Guide and Intervention (continued)Solving Logarithmic Equations and Inequalities

Solving Logarithmic 'nequalities

Property of Inequality for Logarithmic Functions

If b ( 1, x ( ), and logb  x ( y , then x ( b y "

If b ( 1, x ( ), and logb  x * y , then ) * x * b y

"If b ( 1, then logb  x ( logb  y if and on! if x ( y ,

and logb  x * logb  y if and on! if x * y "

Example 1: Solve log5  (! x  3 ) 3.

log5  (4 x ! 3 % 3 Ori#ina e$uation

$ % 4 x ! 3 % 53

Propert! of Ine$uait!

3 % 4 x % 12# + 3 %impif!"

3

4

 % x % 32 %impif!"

The solution set is { x|34  

8/16/2019 15 - Solving Logarithmic Equations and Inequalities

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Exercises

Solve each inequalit*.

1. log2  2 x 2 2. log5   x 2

3. log2  (3 x + 1 % 4 !. log4  2 x −

12

". log3  ( x + 3 % 3 #. log27  " x 2

3

7. log10  # x % log10  3$ $. log10   x % log10  (2 x ! 4

%. log10  3 x % log10  (7 x ! 8 1&. log2  (8 x + # log2  ( x ! 18

11. log10  (3 x + 7 % log10  (7 x ! 3 12. log2  (3 x ! 4 % log2  (2 x + 7