University:
University of North Georgia
Course:
MATH 1113 | Precalculus
Academic year:

2021
Views:

222

Pages:

2
Author:

5CelestialChronicle

Algebra 2 Honors Properties of Logarithms Logs Notes 3 (10.5 Part 1) Warm Up a. Show that log₄81 - log₄3 - log₄27 = 3 by simplifying the three logarithms. 4 - 1 - 3 = 3 b. Show that log₂25 - log₂125 = log₂(1/5) by simplifying the three logarithms. 2 - 3 = -1 c. State a generalization based on parts (a) and (b). logₐb - logₐc = logₐ(b/c) Laws of Logarithms Let b be the base of a logarithmic function (b > 0, b ≠ 1). Let M and N be positive numbers. logₐ(MN) = logₐM + logₐN logₐ(M/N) = logₐM - logₐN logₐ(M^k) = k * logₐM EXAMPLE 1: Applying Properties of Logarithms to Expand Apply the Properties of Logarithms to expand each of the following logarithmic expressions. a. log₂(a²b) 1 log₂a² + log₂b = 2log₂a + log₂b b. log₁₀(100a³b½) log₁₀100 + log₁₀a³ + log₁₀b½ = 2 + 3log₁₀a + ½log₁₀b c. log₅√N³ log₅N^(3/2) = (3/2)log₅N d. log₄(M³/N²√P) log₄M³ - (log₄N² + log₄P^(1/2)) = 3log₄M - 2log₄N - ½log₄P e. log₃(3√a³b⁵) log₃3 + log₃a³ + log₃b⁵ = 1 + ½log₃a + (5/2)log₃b f. log₉(C³ / (9D³)) log₉C³ - log₉(9D³) = 3log₉C - (log₉9 + 3log₉D) = 3log₉C - (1 + 3log₉D) EXAMPLE 2 - Applying Properties of Logarithms to Condense Apply the Properties of Logarithms to condense each of the following logarithmic expressions: a. 1 3log₂m - 2log₂n log₂(m³) - log₂(n²) = log₂(m³/n²) b. 4logM - logN logM⁴ - logN = log(M⁴/N) c. log₂A + 1 + ½log₂E log₂A + log₂2 + ½log₂E = log₂(2A√E) d. ½logᵣ + ½logₛ logᵣ√r + logₛ√s = log(√rs) e. ½(logₓ - logᵧ) logₓ√x - logᵧ√y = log(√x/√y) f. log₃x - 9 + 2log₃y log₃x - log₃19683 + log₃y² = log₃(xy²/19683) EXAMPLE 3 - Logarithmic Equivalencies Given that log₂3 ≈ 1.585 and log₂5 ≈ 2.322, find the following: a. log₂81 log₂3⁴ = 4log₂3 = 4(1.585) = 6.34 b. log₂6 log₂(3*2) = log₂3 + log₂2 = 1.585 + 1 = 2.585 c. log₂75 log₂(5²*3) = 2log₂5 + log₂3 = 2(2.322) + 1.585 = 6.229 d. log₂12 log₂(2²*3) = 2log₂2 + log₂3 = 2(1) + 1.585 = 3.585 Given that log₄2 = 0.60 and log₃2 = 0.48, find the following: a. log₄12 log₄(4*3) = log₄4 + log₄3 = 1 + 0.48 = 1.48 b. log₄¼ log₄(4⁻¹) = -1 c. log₁₀1000 log₁₀10³ = 3 d. log₂√16 log₂(16^(1/3)) = (1/3)log₂16 = (1/3) * 4 = 4/3