University:
High School
Course:
High School Algebra
Academic year:

2021
Views:

448

Pages:

4
Author:

meende6f

Properties of Logarithms In your own words, write the meaning of each vocabulary term. Base: In Exponent Form: The number that is being raised to a power. Exponent: In Log Form: The power to which the base must be raised to obtain the result. Properties of Exponents 2^0 = 1 2^5 / 2^3 = 2^(5-3) = 2^2 (2^5)^3 = 2^(5*3) = 2^15 (2^3)(2^5) = 2^(3+5) = 2^8 Core Concepts Properties of Logarithms Let b, m, and n be positive real numbers with b ≠ 1. Product Property: log_b(mn) = log_bm + log_bn Key: Bases are still the same. Location does matter. Quotient Property: log_b(m/n) = log_bm - log_bn Key: Subtraction Power Property: log_b(m^n) = n * log_bm Key: Multiplication Change-of-Base Formula If a, b, and c are positive real numbers with b ≠ 1 and c ≠ 1, then log_ba = log_ca / log_cb In particular, log_ba = log a / log b and log_ba = ln a / ln b So if you want to find... log_ba ...write it as log a / log b ...both bases are now 10 ...and calculators automatically use log bases of 10 Extra Practice In Exercises 1-4, use log₂5 ≈ 2.322 and log₂12 ≈ 3.585 to evaluate the logarithm. 1. log₂60 = log₂(5 * 12) = log₂5 + log₂12 = 2.322 + 3.585 = 5.907 2. log₂(1/144) = log₂(12)^(-2) = -2 * log₂12 = -2 * 3.585 = -7.17 3. log₂(25/12) = log₂25 - log₂12 = log₂(5)² - log₂12 = 2 * log₂5 - log₂12 = 2 * 2.322 - 3.585 = 4.644 - 3.585 = 1.059 4. log₂720 = log₂(12 * 12 * 5) = log₂12 + log₂12 + log₂5 = 3.585 + 3.585 + 2.322 = 9.492 In Exercises 5-8, expand the logarithmic expression. 5. log(10x) = log 10 + log x = 1 + log x 6. ln(2x^6) = ln 2 + ln x^6 = ln 2 + 6 ln x 7. log₃(x⁴/(3y³)) = log₃x⁴ - log₃(3y³) = 4 log₃x - [log₃3 + log₃y³] = 4 log₃x - 1 - 3 log₃y 8. ln√(3y²) = ln(3y²)^(1/4) = 1/4 [ln(3y²)] = 1/4 [ln 3 + ln y²] = 1/4 ln 3 + 1/4 * 2 ln y = 1/4 ln 3 + 1/2 ln y In Exercises 9-13, condense the logarithmic expression. 9. log₂3 + log₂8 Product = log₂(3 * 8) = log₂24 10. log₅4 - log₅5 Power = log₅4 - log₅(5)² = log₅4 - log₅25 = log₅(4/25) 11. ln 6x + ln 4y Power = ln(6x)³ + ln 4y = ln(6³)x³ + ln 4y = ln(216x³) + ln 4y Product = ln(216x³)(4y) = ln(864x³y) 12. log₂625 - log₂125 + ⅓log₂27 Work left to right = log₂(625/125) + log₂(27)⅓ = log₂5 + log₂3 = log₂(5 * 3) = log₂15 13. -log₆6 - log₆2y + ⅔log₆3x Flip 6 = log₆(6)^(-1) - log₆2y + log₆(3x)^(2/3) = log₆(1/6) - log₆2y + log₆(9x²) = log₆(1/6) - log₆2y + log₆(9x²) since b^1 = b = log₆[(1/6) * (9x²)] reduce = log₆(3x²/2y)