163 Un i t 4 A lgebra : Exponent ia l and Logar i thmic Funct ions

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IT 4

163

Georgia Performance Standards: MM3A3c

Solve Exponential and Logarithmic Inequalities

Goal Solve exponential and logarithmic inequalities.

VocabularyAn exponential inequality in one variable is an inequality that can be written in the form abx 1 k < 0, abx 1 k > 0, abx 1 k ≤ 0, or abx 1 k ≥ 0, where a Þ 0, b > 0, and b Þ 1.

A logarithmic inequality in one variable is an inequality that can be written in the form logb x 1 k < 0, logb x 1 k > 0, logb x 1 k ≤ 0, or logb x 1 k ≥ 0, where b > 0, and b Þ 1.

The same methods used to solve polynomial inequalities in Lesson 2.4 can be used to solve exponential and logarithmic inequalities.

LESSON

4.8

Example 1 Solve an exponential inequality analytically

Solve 4x 1 1 ≥ 32.

Solution

STEP 1 Write and solve the equation obtained by replacing ≥ with 5.

4x 1 1 5 32 Write equation that corresponds to inequality.

log4 4x 1 1 5 log4 32 Take log4 of each side.

x 1 1 5 log4 32 logb bx 5 x

x 5 log4 32 2 1 Subtract 1 from each side.

x 5 log 32}log 4

2 1 Change-of-base formula

x 5 1.5 Use a calculator.

STEP 2 Plot the solution from Step 1 on a number line. Use a solid dot to indicate 1.5 is a solution of the inequality. The solution x 5 1.5 represents the critical x-value of the inequality 4x 1 1 ≥ 32. The critical x-value partitions the number line into two intervals. Test an x-value in each interval to see if it is a solution of the inequality.

�3 �2 �1 0 1

1.5

2 3 4 5 6

Test x 5 0:40 1 1 32

4 ¦ 32

Test x 5 3:43 1 1 32

256 $ 32

The solution consists of all real numbers in the interval [1.5, 1`).

UN

IT 4

164 Georg ia H igh Schoo l Mathemat ics 3

Georgia Performance Standards

MM3A3c Solve polynomial, exponential, and logarithmic inequalities analytically, graphically, and using appropriate technology. Represent solution sets of inequalities using interval notation.

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Example 2 Solve an exponential inequality using technology

Car Value Your family purchases a new car for $25,000. Its value depreciates by 12% each year. During what interval of time does the car’s value exceed $16,000?

Solution

Let y represent the value of the car (in dollars) x years after it is purchased. A function relating x and y is:

y 5 25,000(1 2 0.12)x Original function

y 5 25,000(0.88)x Simplify.

To fi nd the values of x for which y exceeds 16,000, solve the inequality 25,000(0.88)x > 16,000.

METHOD 1 Use a table.

STEP 1 Enter the function y 5 25,000(0.88)x into a X Y13.2 166073.3 16396

3.5 159823.6 157793.7 15779X=3.4

3.4 16188

graphing calculator. Set up a table to display the x-values starting at 0 and increasing in increments of 0.1.

STEP 2 Use the table feature to create a table of values. Scrolling through the table shows that y > 16,000 when 0 ≤ x ≤ 3.4.

The car’s value exceeds $16,000 for about the fi rst 3.4 years after it is purchased.

METHOD 2 Use a graph.

STEP 1 Graph y 5 25,000(0.88)x and y 5 16,000

IntersectionX=3.4911627 Y=16000

in the same viewing window. Set the viewing window to show 0 ≤ x ≤ 8 and 0 ≤ y ≤ 25,000.

STEP 2 Use the intersect feature to determine where the graphs intersect. The graphs intersect when x ø 3.49.

The graph of y 5 25,000(0.88)x is above the graph of y 5 16,000 when 0 ≤ x ≤ 3.49. So, the car’s value exceeds $16,000 for about the fi rst 3.5 years after it is purchased.

Guided Practice for Examples 1 and 2

Solve the exponential inequality.

1. 5x 2 4 ≥ 625 2. 32x 2 1 ≤ 4 3. 73x 2 4 < 18

4. Car Value In Example 2, during what interval of time does the car’s value fall below $10,000?

165 Un i t 4 A lgebra : Exponent ia l and Logar i thmic Funct ions

UN

IT 4

UN

IT 4

Example 3 Solve a logarithmic inequality analytically

Solve the logarithmic inequality.

a. log5 x < 2 b. log4 x 1 8 ≥ 11

Solution

a. STEP 1 Write and solve the equation obtained by replacing < with 5.

log5 x 5 2 Write equation that corresponds to inequality.

5log5 x 5 52 Exponentiate each side using base 5.

x 5 25 blogb x 5 x

STEP 2 Plot the solution from Step 1 using an open dot. The value x must be a positive number, so the number line should show only positive numbers. The solution x 5 25 represents the critical x-value of the inequality log5 x < 2. Test an x-value in each interval to see if it is a solution of the inequality.

0 15105 20 25 30 35 40 45

Test x 5 5:log5 5 , 2

1 , 2

Test x 5 40:log5 40 , 2

2.29 ² 2

The solution consists of all real numbers in the interval (0, 25).

b. STEP 1 Write and solve the equation obtained by replacing ≥ with 5.

log4 x 1 8 5 11 Write equation that corresponds to inequality.

log4 x 5 3 Subtract 8 from each side.

4log4 x 5 43 Exponentiate each side using base 4.

x 5 64 blogb x 5 x

STEP 2 Plot the solution from Step 1 using a solid dot. The solution x 5 64 represents the critical x-value of the inequality log4 x 1 8 ≥ 11. Test an x-value in each interval to see if it is a solution of the inequality.

4832160 64 80 96 112 128 144

Test x 5 128:log4 128 1 8 $ 11

11.5 $ 11

Test x 5 16:log4 16 1 8 $ 11

10 ¦ 11

The solution consists of all real numbers in the interval [64, 1`).

Guided Practice for Example 3

Solve the logarithmic inequality.

5. log3 x 2 3 > 1 6. log6 2x 1 7 < 10 7. log9(x 2 8) ≥ 3}2

166 Georg ia H igh Schoo l Mathemat ics 3

UN

IT 4

Example 4 Solve a logarithmic inequality using technology

Solve log3 x ≤ 2.

Solution

METHOD 1 Use a table.

STEP 1 Enter the function y 5 log3 x into a Y1= log(X)/log(3)Y2Y3Y4Y5=Y6=Y7=

===

graphing calculator as y 5 log x}log 3

.

STEP 2 Use the table feature to create a table X Y16 1.6309

8 1.8928

10 2.0959X=9

7 1.7712

9 2

of values. Identify the interval for which y ≤ 2. These x-values can be represented by the interval (0, 9].

Make sure that the x-values are reasonable and in the domain of the function (x > 0).

The solution of log3 x ≤ 2 is (0, 9].

METHOD 2 Use a graph.

STEP 1 Graph y 5 log3 x and y 5 2 in the same

IntersectionX=9 Y=2

viewing window.

STEP 2 Using the intersect feature, you can determine that the graphs intersect when x 5 9.

The graph of y 5 log3 x is on or below the graph of y 5 2 during the x interval (0, 9].

So, the solution of log3 x ≤ 2 is (0, 9].

Guided Practice for Example 4

Use a graphing calculator to solve the logarithmic inequality.

8. log2 x < 3

9. log5(x 2 2) ≥ 1}3

10. log4 3x > 2}5