Concept Byte: Exponential and Logarithmic Inequalities

For Use With Lesson 7-6

EXTENSION

You can use the graphing and table capabilities of your calculator to solve problems involving exponential and logarithmic inequalities.

Example 1

Solve 2 ( 3 ) x + 4 > 10 2 . open 3 close super x plus 4 end super . greater than 10  using a graph.

Step 1 Define Y1 and Y2.

The solution is x > − 2.535 . x greater than negative , 2.535 , .

Step 2 Make a graph and find the point of intersection.

Step 3 Identify the x-values that make the inequality true.

Exercises

Solve each inequality using a graph.

1. 4 ( 3 ) x + 1 > 6 4 . open 3 close super x plus 1 end super . greater than 6
2. log x + 3 log ( x − 1 ) < 4 log x plus 3 log open x minus 1 close less than 4
3. 3 ( 2 ) x + 2 ≥ 5 3 . open 2 close super x plus 2 end super . greater than or equal to 5
4. x + 1 < 12 log x x plus 1 less than 12 log x
5. 2 ( 3 ) x − 4 > 7 2 . open 3 close super x minus 4 end super . greater than 7
6. log x + 2 log ( x − 1 ) < 1 log x plus 2 log open x minus 1 close less than 1
7. 4 ( 2 ) x − 1 ≤ 5 4 . open 2 close super x minus 1 end super . less than or equal to 5
8. 2 log x + 4 log ( x + 3 ) > 3 2 log x plus 4 log open x plus 3 close greater than 3
9. 5 ( 4 ) x − 1 < 2 5 . open 4 close super x minus 1 end super . less than 2
10. Bacteria Growth Scientists are growing bacteria in a laboratory. They start with a known population of bacteria and measure how long it takes the population to double.
    1. Write an exponential function that models the population in Sample A as a function of time in hours.
    2. Write an exponential function that models the population in Sample B as a function of time in hours.
    3. Write an inequality that models the population in Sample B overtaking the population in Sample A.
    4. Use a graphing calculator to solve the inequality in part (c).
11. Writing Describe the solution sets to the inequality x + c < log x x plus c less than log x  as c varies over the real numbers.

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