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