Chapter 6 Exponential and Logarithmic Functions and Applications Section 6.4

Section 6.4 Exponential and Logarithmic Equations Solving Exponential Equations Exponential Equality Property Base 10 or Base e Graphically Fundamental Properties of Logarithms Solving Exponential Equations Using Properties of Logarithms Solving Logarithmic Equations Solving Literal Equations Involving Exponential and Logarithmic Equations Change of Base Formula Applications

For b > 0 and b  1, if bm = bn, then m = n. Exponential Equality Property: For b > 0 and b  1, if bm = bn, then m = n. *Solving Exponential Equations with the Exponential Equality Property: Express each side of the equation as a power of the same base. Apply the exponential equality property to equate the exponents. 3. Solve for the variable. 4. Check your solution. *Also referred to as solving by "equating the bases" or "relating the bases."

Use the exponential equality property to solve 292x + 4 = 162. Step 1. Express each side of the equation as a power of the same base. First, we divide both sides of 292x + 4 = 162 by “2.” 92x + 4 = 81 Next, we express each side of the equation as a power of “3.” (32)2x + 4 = 34 34x + 8 = 34 Make sure to distribute correctly: 2(2x + 4) = 4x + 8 Steps 2-3. Apply the exponential equality property to equate the exponents. Solve for the variable. 4x + 8 = 4 x = –1 Step 4. Check your solution. 29[2(–1) + 4] = 162  292 = 162  2(81) = 192 True

Use the exponential equality property to solve Check:

2. Convert the equation to logarithmic form. Solving Exponential Equations with Base 10 or Base e. Isolate the power (the term containing the variable exponent) on one side of the equation. If necessary, divide both sides of the equation by any coefficient of the power term. 2. Convert the equation to logarithmic form. 3. Solve for the variable. Use a calculator if necessary. 4. Check your solution.

Solve the exponential equation 2(–6 + 102x) = 16. 26 Solve the exponential equation 2(–6 + 102x) = 16.26. Round your answer to 4 decimal places as needed. Step 1. Isolate the power. First, divide both sides of the equation by “2.” Then, add “6” to both sides of the equation. –6 + 102x = 8.13 102x = 14.13 Step 2. Convert the equation to logarithmic form. log1014.13 = 2x or log 14.13 = 2x Step 3. Solve for the variable. x = log 14.13 2 x  0.5751 Step 4. Check your solution. 2[–6 + 10(2 0.5751)] = 16.26  2(8.1319) = 16.2638  16.26

Solve the exponential equation 25. 6 = –2 + 3. 1e–0. 15x Solve the exponential equation 25.6 = –2 + 3.1e–0.15x. Round your answer to 4 decimal places as needed. 27.6 = 3.1e–0.15x 3.1 3.1 8.9032 = e–0.15x loge 8.9032 = –0.15x or ln 8.9032 = –0.15x x = ln 8.9032 –0.15 x  –14.5761 You can check the answer.

Can use intersection method or x-intercept method. Solving Exponential Equations Graphically Use your graphing calculator to solve 2x + 5 = 32x + 6 – 1 graphically. Can use intersection method or x-intercept method. Let Y1 = 2x + 5 and Y2 = 32x + 6 – 1. We will use the window [–6, 6, 1] by [– 6, 12, 1] and find the intersection. The answer to the given equation is x = –2. Checking: 2(–2 + 5) = 3[2(–2) + 6] – 1 8 = 8

Note: These properties apply to natural logarithms as well. Recall the Basic Properties of Logarithms: For b > 0 and b  1, logb 1 = 0, logb b = 1, logb bx = x, and Fundamental Properties of Logarithms For positive real numbers M, N, and b, b  1, and any real number k: Product Property: Quotient Property: Power Property: Note: These properties apply to natural logarithms as well.

= log3 (47) = log3(4) + log3(7) = 1.2619 + 1.7712 = 3.0331 Use the fact that log3 4 = 1.2619 and log3 7 = 1.7712 and the properties of logarithms to estimate the value of the following expressions. Round your answers to 4 decimal places as needed. a. log3 28 = log3 (47) = log3(4) + log3(7) = 1.2619 + 1.7712 = 3.0331 b. log3 (1.75) = log3 (7/4) = log3(7) – log3(4) = 1.7712 – 1.2619 = 0.5093 c.

a. 5 log x + log 9 = log x5 + log 9 = log (9x5) Use the properties of logarithms to rewrite each expression as a single logarithm. a. 5 log x + log 9 = log x5 + log 9 = log (9x5) b. log (x + 3) – log (x2 – 9) c. or

Expand the given expression in terms of simpler logarithms Expand the given expression in terms of simpler logarithms. Assume that all variable expressions are positive real numbers.

Solving Exponential Equations Using Properties of Logarithms 1. Isolate the power on one side of the equation. 2. Take the logarithm of both sides of the equation; may take either the common logarithm (base 10) or a natural logarithm (base e). 3. Apply the power property of logarithms to simplify (that is, "bring down" the variable exponent to the front). 4. Solve for the variable. 5. Check your solution.

Step 1: Isolate the power on one side of the equation. 4x = 12 Solve the exponential equation –7 + 4x = 5. Round your answer to 4 decimal places. Step 1: Isolate the power on one side of the equation. 4x = 12 Step 2: Take the logarithm of both sides of the equation. log 4x = log 12 Step 3: Apply the power property of logarithms. x log 4 = log 12 Step 4: Solve for the variable. 5. Check your solution. –7 + 4(1.7925)  5

Solve the exponential equation 27ex – 2 = 7 Solve the exponential equation 27ex – 2 = 7. Round your answer to 4 decimal places. ex – 2 = 0.2593 ln ex – 2 = ln 0.2593 (x – 2) ln e = ln 0.2593 (x – 2) (1) = ln 0.2593 (Recall ln e is equivalent to loge e) x = ln 0.2593 + 2 x  0.6502 Checking: 27e(0.6502 – 2)  7

For positive real numbers m, n, and b, b  1, Logarithmic Equality Property: For positive real numbers m, n, and b, b  1, if logb m = logb n, then m = n. Solving Logarithmic Equations: 1. Isolate the logarithmic expression on one side of the equation. If needed, apply the properties of logarithms to combine all logarithms as a single logarithm. 2. Convert the logarithmic equation to an exponential equation. 3. Solve for the variable. 4. Check for possible extraneous solutions.

Solve the logarithmic equation 6 log 4x = 18. Step1. Isolate the logarithmic expression on one side of the equation. log 4x = 3 Step 2: Convert the logarithmic equation to an exponential 103 = 4x Step 3: Solve for the variable. 4x = 1000 x = 250 Step 4: Check for possible extraneous solutions. 6 log [4(250)] = 18 18 = 18

Solve the logarithmic equation We discard x = –10/3 (it is not in the domain). Thus, x = 10. You can verify the solution.

Solve the equation ln x + ln (x + 3) = ln (x + 15). ln [x(x + 3)] = ln (x + 15) ln (x2 + 3x) = ln (x + 15) x2 + 3x = x + 15  Logarithmic equality x2 + 3x – x – 15 = 0 x2 + 2x – 15 = 0 (x + 5)(x – 3) = 0 x = –5 or x = 3 We discard x = –5 (it is not in the domain). Thus, x = 3. You can verify the solution.

Recall ln (2x – 1) = 3 is equivalent to loge(2x – 1) = 3 e3 = 2x – 1 Solve the equation ln (2x – 1) = 3 and approximate your answer to 4 decimal places. Recall ln (2x – 1) = 3 is equivalent to loge(2x – 1) = 3 e3 = 2x – 1 2x = e3 + 1 2x = 21.085537 x  10.5428

Solve for y: Solving Literal Equations Involving Exponential or Logarithmic Equations Solve for y:

Change of Base Formula For positive real numbers M, a, and b, a  1, b  1, We may convert any given base into either base 10 or base e. Example: Use the change of base formula to evaluate log2 5, and graph y = log2 x by applying the change of base formula. or Graph of y = log2 x:

The function P(t) = 89.371(3.2)t models the number of digital 3D screens worldwide for t number of years after 2005. Using this model, estimate when the number of digital 3D screens worldwide reached approximately 9,000. Solve algebraically and round your answer to the nearest whole number. Source: mpaa.org. The number of digital 3D screens reached approximately 9,000 in 2009.

The number of prohibited firearms intercepted by the Transportation Security Administration (TSA) at U.S. airport screenings from 2005 to 2009 can be modeled by the function f(x) = 2391.893 – 931.691 ln x, where x = 1 represents 2005, x = 2 is 2006, and so on. Use this model to estimate when the TSA intercepted approximately 892 firearms. Round your answer to the nearest whole number. Sources: www.aaa.com; www.safecarguide.com   The TSA intercepted approximately 892 firearms in 2009.

Using your textbook, practice the problems assigned by your instructor to review the concepts from Section 6.4.