Math Chapter 3 Exponential and Logarithmic Functions

Chapter 2: Exponential and Logarithmic Functions 1. Evaluate the following exponential expressions: a. 3^(-3) * 3^(10) / 3^2 = 3^(-3+10-2) = 3^5 = 243 b. (4^(-2) + 4^0)^(-1) = (1/16 + 1)^(-1) = (17/16)^(-1) = 16/17 c. √(9 * 5^4) / √(3^(-2)) = √(3^2 * 5^4) / 3^(-1) = 3 * 5^2 * 3 = 729 d. 2^(-4) + (2^(-4) + 2^2)^2 = 1/16 + (1/16 + 4)^2 = 1/16 + (65/16)^2 = 265/16 e. √((3^(-1))^8) + 8^(-4/3) = √(3^(-8)) + (2^3)^(-4/3) = 3^(-4) + 2^(-4) = 1/81 + 1/16 f. 6x^(-2) + 2x^(-1) + 6x = 6/x^2 + 2/x + 6x = (6 + 2x + 6x^3) / x^2 2. Find the exact value (without a calculator): a. log₂16 2^x = 16 x = 4 b. log₂₇3 3^x = 27 3^x = 3^3 x = 3 c. 5^(log₅64) 5^(log₅64) = 64 3. Given log 40 = 1.6021, using rules of logarithms, find the value of the following: a. log 4 log 4 = log(40/10) = log 40 - log 10 log 4 ≈ 1.6021 - 1 = 0.6021 b. log 1000 log 1000 = log(10 * 100) = log 10 + log 100 log 1000 ≈ 1 + 2 = 3 4. Expand. a. log₃(√M / N³√P) log₃M^(1/2) - log₃N³ - log₃P^(1/2) (1/2)log₃M - 3log₃N - (1/2)log₃P b. ln(5e^(x+2)) ln5 + ln(e^(x+2)) ln5 + (x+2)lne ln5 + (x+2) c. ln(√(x+3)/x√y) ln(√(x+3)) - ln(x√y) (1/2)ln(x+3) - ln(x) - ln(√y) (1/2)ln(x+3) - ln(x) - (1/2)ln(y) 5. Condense. a. ln 3 + (1/3)ln(4-x²) - lnx ln(3 * (4-x²)^(1/3) / x) b. ln(x²) - (1/2)ln(x+2) - 3lnx ln(x² / √(x+2) * x³) ln(x² / x^(7/2)(x+2)) Solve the equation. Round answers to the nearest thousandth where necessary. a. 3^(8-x) = 27^(2-x) 3^(8-x) = 3^(3(2-x)) 8-x = 3(2-x) 8-x = 6-3x 2x = -2 x = -1 b. 5^(x+1) = 2^(3x+6) (x+1)log5 = (3x+6)log2 0.699(x+1) = 0.301(3x+6) 0.699x + 0.699 = 0.903x + 1.806 -0.204x = 1.107 x ≈ -5.421 c. e^(2x+1) = 2 2x+1 = ln2 2x = ln2 - 1 x = (ln2 - 1)/2 ≈ -0.307 d. 2^(x+2) = 35 x+2 = log₂35 x = log₂35 - 2 ≈ 3.459 e. 10^(4-x) = 4 4-x = log10(4) x = 4 - log10(4) ≈ 3.398 f. 5^(x-3) = 2 x-3 = log₅2 x = log₅2 + 3 ≈ 3.431 g. 2^(x+1) = 16 2^(x+1) = 2^4 x+1 = 4 x = 3 h. (e^x)² - 3e^x + 2 = 0 (e^x - 1)(e^x - 2) = 0 e^x = 1 or e^x = 2 x = 0 or x = ln2 ≈ 0.693 i. log₅(5√5) = x log₅(5^(3/2)) = x (3/2)log₅5 = x x = 3/2 j. 2^(x-1) = 4 2^(x-1) = 2^2 x-1 = 2 x = 3 k. √(3^x) = 81 3^(x/2) = 3^4 x/2 = 4 x = 8 l. ln x = 4 x = e^4 ≈ 54.598 m. log₂(x²+5x) = log₂(4x+2) x²+5x = 4x+2 x² + x - 2 = 0 (x+2)(x-1) = 0 x = -2 or x = 1 n. log₂(3) + log₂(x) = log₂(x+1) log₂(3x) = log₂(x+1) 3x = x+1 2x = 1 x = 1/2 o. log₂(7-x) - log₂(x+1) = 1 log₂((7-x)/(x+1)) = 1 (7-x)/(x+1) = 2 7-x = 2x+2 -3x = -5 x = 5/3 p. log₂(x-2) + log₂(x+5) = 1 log₂((x-2)(x+5)) = 1 (x-2)(x+5) = 2 x² + 3x - 10 = 2 x² + 3x - 12 = 0 x = (-3 ± √(9+48))/2 x = (-3 ± √57)/2 7. Graph the following and label the x-intercept, y-intercept and asymptotes. Also, find the domain, range, and the interval over which it is increasing or decreasing. Lastly, list the transformations from the parent function to g(x).   a. g(x) = 3^(x-1) - 5   [Image of the graph of g(x) = 3^(x-1) - 5] D of g(x): (-∞, ∞) R of g(x): (-5, ∞) Asymptote of g(x): y = -5 g(x): Inc./Dec.: (-∞, ∞) Transformations from parent to g(x): Right 1 Down 5 b. g(x) = -3^(x+3) + 4 [Image of the graph of g(x) = -3^(x+3) + 4] D of g(x): (-∞, ∞) R of g(x): (-∞, 4) Asymptote of g(x): y = 4 g(x): Inc./Dec.: (-∞, ∞) Transformations from parent to g(x): Left 3 Reflect over x-axis Up 4 c. g(x) = -log₂(x-3) [Image of the graph of g(x) = -log₂(x-3)] D of g(x): (3, ∞) R of g(x): (-∞, ∞) Asymptote of g(x): x = 3 g(x): Inc./Dec.: (3, ∞) Transformations from parent to g(x): Right 3 Reflect over x-axis d. g(x) = log₂(-x) + 3 [Image of the graph of g(x) = log₂(-x) + 3] D of g(x): (-∞, 0) R of g(x): (-∞, ∞) Asymptote of g(x): x = 0 g(x): Inc./Dec.: (-∞, 0) Transformations from parent to g(x): Reflect over y-axis Up 3