1. Solving Logarithmic Equations and Inequalities
LESSON 7–4
Solving Logarithmic Equations and Inequalities

2. Write 4–3 = in logarithmic form.__ 1 64
A. log–3 4 =
B. log– = 4
C. log = –3
D. log4 –3 = __ 1 64
5-Minute Check 1

3. Write log6 216 = 3 in exponential form.
B. 36 = 216 C. D.
5-Minute Check 2

4. Graph f(x) = 2 log2 x. C. D.
A. ans
B. ans
5-Minute Check 3

5. Graph f(x) = log3 (x – 4). A. B. C. D.
5-Minute Check 4

6. A. B. C. D.
5-Minute Check 5

7. Mathematical Processes A2.1(F), Also addresses A2.1(E).
Targeted TEKS
A2.5(B) Formulate exponential and logarithmic equations that model real-world situations, including exponential relationships written in recursive notation.
A2.5(D) Solve exponential equations of the form y = abx where a is a nonzero real number and b is greater than zero and not equal to one and single logarithmic equations having real solutions.
Also addresses A2.5(E).
Mathematical Processes
A2.1(F), Also addresses A2.1(E).
TEKS

8. You evaluated logarithmic expressions.
Solve logarithmic equations.
Solve logarithmic inequalities.
Then/Now

9. logarithmic inequality
logarithmic equation
logarithmic inequality
Vocabulary

10. Definition of logarithm
Solve a Logarithmic Equation
Solve
Original equation
Definition of logarithm
8 = 23
Power of a Power
Answer: x = 16
Example 1

11. Solve A.
B. n = 3
C. n = 9 D.
n =
Example 1

12. Concept

13. You need to find x for the logarithmic equation.
Solve a Logarithmic Equation
Solve log4 x 2 = log4 (–6x – 8).
A B C. –4, – D. no solutions
Read the Item
You need to find x for the logarithmic equation.
Solve the Item
log4 x 2 = log4 (–6x – 8) Original equation
x 2 = (–6x – 8) Property of Equality for Logarithmic Functions
Example 2

14. x 2 + 6x + 8 = 0 Subtract (–6x – 8) from each side.
Solve a Logarithmic Equation
x 2 + 6x + 8 = 0 Subtract (–6x – 8) from each side.
(x + 4)(x + 2) = 0 Factor.
x + 4 = 0 or x + 2 = 0 Zero Product Property
x = – x = –2 Solve each equation.
Example 2

15. Substitute each value into the original equation.
Solve a Logarithmic Equation
Check
Substitute each value into the original equation.
x = –4 ?
log4 (–4)2 = log4 [–6(–4) – 8)]
log4 16 = log4 16 
x = –2 ?
log4 (–2)2 = log4 [–6(–2) – 8)]
log4 4 = log4 4 
Answer: The solutions are x = –4 and x = –2. The answer is C.
Example 2

16. Solve log4 x 2 = log4 (x + 20). A. 5 and –4 B. –2 and 10 C. 2 and –10
D. no solutions
Example 2