3.4 Quick Review Express In 56 in terms of ln 2 and ln 7. A. ln 23 + ln 7 B. ln 7 – 3 ln 2 C. 3 ln 2 – ln 7 D. 3 ln 2 + ln 7

3.4 Quick Review Express In 56 in terms of ln 2 and ln 7. A. ln 23 + ln 7 B. ln 7 – 3 ln 2 C. 3 ln 2 – ln 7 D. 3 ln 2 + ln 7

3.4 Quick Review Expand . A. B. C. D.

3.4 Quick Review Expand . A. B. C. D.

3.4 Quick Review Condense . A. AAAA B. BBBB C. CCCC D. DDDD

3.4 Quick Review Condense . A. AAAA B. BBBB C. CCCC D. DDDD

3.4 Quick Review Estimate log3 7 to the nearest whole number. A. 0 D. 3

3.4 Quick Review Estimate log3 7 to the nearest whole number. A. 0 D. 3

Key Concept 1

4x + 2 = 16x – 3 Original equation 4x + 2 = (42)x – 3 42 = 16 Solve Exponential Equations Using One-to-One Property A. Solve 4x + 2 = 16x – 3. 4x + 2 = 16x – 3 Original equation 4x + 2 = (42)x – 3 42 = 16 4x + 2 = 42x – 6 Power of a Power x + 2 = 2x – 6 One-to-One Property 2 = x – 6 Subtract x from each side. 8 = x Add 6 to each side. Answer: Example 1

B. Solve . Original equation Power of a Power Solve Exponential Equations Using One-to-One Property B. Solve . Original equation Power of a Power Example 1

One-to-One Property. Answer: Solve Exponential Equations Using One-to-One Property One-to-One Property. Answer: Example 1

Solve 25x + 2 = 54x. A. 1 B. C. 2 D. –2 Example 1

Solve 25x + 2 = 54x. A. 1 B. C. 2 D. –2 Example 1

A. Solve 2 ln x = 18. Round to the nearest hundredth. Solve Logarithmic Equations Using One-to-One Property A. Solve 2 ln x = 18. Round to the nearest hundredth. Method 1 Use exponentiation. 2 ln x = 18 Original equation ln x = 9 Divide each side by 2. eln x = e9 Exponentiate each side. x = e9 Inverse Property x ≈ 8103.08 Use a calculator. Example 2

Method 2 Write in exponential form. 2 ln x = 18 Original equation Solve Logarithmic Equations Using One-to-One Property Method 2 Write in exponential form. 2 ln x = 18 Original equation ln x = 9 Divide each side by 2. x = e9 Write in exponential form. x ≈ 8103.08 Use a calculator. Answer: Example 2

B. Solve 7 – 3 log 10x = 13. Round to the nearest hundredth. Solve Logarithmic Equations Using One-to-One Property B. Solve 7 – 3 log 10x = 13. Round to the nearest hundredth. 7 – 3 log 10x = 13 Original equation –3 log 10x = 6 Subtract 7 from each side. log 10x = –2 Divide each side by –3. 10–2 =10x Write in exponential form. 10–3 = x Divide each side by 10. = x = 10–3. Answer: Example 2

C. Solve log5 x 4 = 20. Round to the nearest hundredth. Solve Logarithmic Equations Using One-to-One Property C. Solve log5 x 4 = 20. Round to the nearest hundredth. log5x4 = 20 Original equation 4 log5x = 20 Power Property log5x = 5 Divide each side by 4. x = 55 Write in exponential form x = 3125 Simplify. Answer: Example 2

Solve 2 log2x 3 = 18. A. 81 B. 27 C. 9 D. 8 Example 2

Solve 2 log2x 3 = 18. A. 81 B. 27 C. 9 D. 8 Example 2

Key Concept 2

A. Solve log2 5 = log2 10 – log2 (x – 4). Solve Exponential Equations Using One-to-One Property A. Solve log2 5 = log2 10 – log2 (x – 4). log25 = log210 – log2(x – 4) Original equation log25 = Quotient Property 5 = One-to-One Property 5x – 20 = 10 Multiply each side by x – 4.5 5x = 30 Add 20 to each side. x = 6 Divide each side by 5. Answer: Example 3

log5(x2 + x) = log520 Original equation Solve Exponential Equations Using One-to-One Property B. Solve log5 (x2 + x) = log5 20. log5(x2 + x) = log520 Original equation x2 + x = 20 One-to-One Property x2 + x – 20 = 0 Subtract 20 from each side. (x – 4)(x + 5) = 0 Factor x 2 + x – 20 into linear factors. x = –5 or 4 Solve for x. Check this solution. Answer: Example 3

Solve log315 = log3x + log3(x – 2). A. 5 B. –3 C. –3, 5 D. no solution Example 3

Solve log315 = log3x + log3(x – 2). A. 5 B. –3 C. –3, 5 D. no solution Example 3

A. Solve 3x = 7. Round to the nearest hundredth. Solve Exponential Equations A. Solve 3x = 7. Round to the nearest hundredth. 3x = 7 Original equation log 3x = log 7 Take the common logarithm of each side. x log 3 = log 7 Power Property x = or about 1.77 Divide each side by log 3 and use a calculator. Answer: Example 4

B. Solve e2x + 1 = 8. Round to the nearest hundredth. Solve Exponential Equations B. Solve e2x + 1 = 8. Round to the nearest hundredth. e2x + 1 = 8 Original equation ln e2x + 1 = ln 8 Take the natural logarithm of each side. 2x + 1 = ln 8 Inverse Property x = or about 0.54 Solve for x and use a calculator. Answer: Example 4

Solve 4x = 9. Round to the nearest hundredth. B. 1.58 C. 2.25 D. 0.44 Example 4

Solve 4x = 9. Round to the nearest hundredth. B. 1.58 C. 2.25 D. 0.44 Example 4

Solve 36x – 3 = 24 – 4x. Round to the nearest hundredth. Solve in Logarithmic Terms Solve 36x – 3 = 24 – 4x. Round to the nearest hundredth. 36x – 3 = 24 – 4x Original equation ln 36x – 3 = ln 24 – 4x Take the natural logarithm of each side. (6x – 3) ln 3 = (4 – 4x) ln 2 Power Property 6x ln 3 – 3 ln3 = 4 ln 2 – 4x ln 2 Distributive Property 6x ln 3 + 4x ln 2 = 4 ln 2 + 3 ln3 Isolate the variable on the left side of the equation. x(6 ln 3 + 4 ln 2) = 4 ln 2 + 3 ln3 Distributive Property Example 5

x(ln 36 + ln 24) = ln 24 + ln 33 Power Property Solve in Logarithmic Terms x(ln 36 + ln 24) = ln 24 + ln 33 Power Property x ln [36(24)] = ln [24(33)] Product Property x ln 11,664 = ln 432 36(24) = 11,664 and 24(33) = 432 x = Divide each side by ln 11,664. x ≈ 0.65 Use a calculator. Answer: Example 5

Solve 4x + 2 = 32 – x. Round to the nearest hundredth. B. 1.08 C. 0.68 D. –0.23 Example 5

Solve 4x + 2 = 32 – x. Round to the nearest hundredth. B. 1.08 C. 0.68 D. –0.23 Example 5

e2x – ex – 2 = 0 Original equation Solve Exponential Equations in Quadratic Form Solve e2x – ex – 2 = 0. e2x – ex – 2 = 0 Original equation u2 – u – 2 = 0 Write in quadratic form by letting u = ex. (u – 2)(u + 1) = 0 Factor. u = 2 or u = –1 Zero Product Property ex = 2 ex = –1 Replace u with ex. Example 6

ln ex = ln 2 ln ex = ln (–1) Take the natural logarithm of each side. Solve Exponential Equations in Quadratic Form ln ex = ln 2 ln ex = ln (–1) Take the natural logarithm of each side. x = ln 2 x = ln (–1) Inverse Property or about 0.69 The only solution is x = ln 2 because ln (–1) is extraneous. Answer: Example 6

e2x – ex – 2 = 0 Original equation Solve Exponential Equations in Quadratic Form Check e2x – ex – 2 = 0 Original equation e2(ln 2) – eln 2 – 2 = 0 Replace x with ln 2. eln 22– eln 2 – 2 = 0 Power Property 22 – 2 – 2 = 0 Inverse Property 0 = 0 Simplify. Example 6

Solve e2x + ex – 12 = 0. A. ln 3 B. ln 3, ln 4 C. ln 4 D. ln 3, ln (–4) Example 6

Solve e2x + ex – 12 = 0. A. ln 3 B. ln 3, ln 4 C. ln 4 D. ln 3, ln (–4) Example 6

Solve log x + log (x – 3) = log 28. Solve Logarithmic Equations Solve log x + log (x – 3) = log 28. log x + log (x – 3) = log 28 Original equation log x(x – 3) = log 28 Product Property log (x 2 – 3x) = log 28 Simplify. x 2 – 3x = 28 One-to-One Property x 2 – 3x – 28 = 0 Subtract 28 from each side. (x – 7)(x + 4) = 0 Factor. x = 7 or x = – 4 Zero Product Property Example 7

The only solution is x = 7 because –4 is an extraneous solution. Solve Logarithmic Equations The only solution is x = 7 because –4 is an extraneous solution. Answer: Example 7

Solve ln x + ln (5 – x) = ln 6. A. 2 B. 3 C. 2, 3 D. –2, –3 Example 7

Solve ln x + ln (5 – x) = ln 6. A. 2 B. 3 C. 2, 3 D. –2, –3 Example 7

Solve log (3x – 4) = 1 + log (2x + 3). Check for Extraneous Solutions Solve log (3x – 4) = 1 + log (2x + 3). log (3x – 4) = 1 + log (2x + 3) Original Equation log (3x – 4) – log (2x + 3) = 1 Subtract log (2x + 3) from each side. = 1 Quotient Property = log 101 Inverse Property = log 10 101 = 10 Example 8

3x – 4 = 10(2x + 3) Multiply each side by 2x + 3. Check for Extraneous Solutions = 10 One-to-One Property 3x – 4 = 10(2x + 3) Multiply each side by 2x + 3. 3x – 4 = 20x + 30 Distributive Property –17x = 34 Subtract 20x and add 4 to each side. x = –2 Divide each side by –17. Example 8

Section 3.4 Homework p. 196; 5-13 odd, 23, 24, 27, 33, 35, 39, 41, 51-53, 61-63, 71