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

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

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

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

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

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

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

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

logarithmic inequality logarithmic equation logarithmic inequality Vocabulary

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

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

Concept

You need to find x for the logarithmic equation. Solve a Logarithmic Equation Solve log4 x 2 = log4 (–6x – 8). A. 4 B. 2 C. –4, –2 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

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 = –4 x = –2 Solve each equation. Example 2

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

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