University:
York University
Course:
MATH 1013 | Applied Calculus
Academic year:

2023
Views:

47

Pages:

3
Author:

Niklas

III. Logarithmic Functions (ch 3.2-3.3) I. DEFINITION: logarithm - the inverse of the exponential function f(x) = 2^x f^(-1)(x) = log_2 x II. SPECIAL LOGARITHMIC BASES common base (log) base 10 natural base (ln) base e (~2.7182818284...) III. LOGARITHMIC PROPERTIES property name mathematical representation change of base log_b a = (log_c a) / (log_c b) zero property log_b 1 = 0 identity property log_b b = 1 product property log_b (mn) = log_b m + log_b n quotient property log_b (m/n) = log_b m - log_b n power property log_b m^n = n * log_b m inverse property log_b (b^x) = x <=> b^(log_b x) = x reciprocal property log_b (1/x) = -log_b x Экспортировать в Таблицы 1. log(9.4) log_3 9 + log_3 4 2 + log_3 4 = 2 + (log 4) / (log 3) ≈ 3.262 2. log_4 8 + log_4 2 log_4 16 = 2 3. log(x^2 y) log x^2 + log y 2 log x + log y IV. SOLVING LOGARITHMIC EQUATIONS: For m, n, b ∈ ℝ, m, n, b > 0, b ≠ 1: log_b m = log_b n ⇒ m = n log_4 (x+5) = log_4 (2x-1) x+5 = 2x-1 note x ∈ (0, ∞] x = 6 ln(x^2+9) = ln(6x) x^2+9 = 6x note x ∈ (0, ∞] x^2-6x+9 = 0 (x-3)^2 = 0 x = 3 V. EXPONENTIAL FUNCTIONS COMPARED TO LOGARITHMIC FUNCTIONS: exponential functions (y = b^x) asymptote: y = 0 domain: (-∞, ∞) range: (0, ∞) logarithmic functions (y = log_b x) asymptote: x = 0 domain: (0, ∞) range: (-∞, ∞) VI. PARENT FUNCTION GRAPH: f(x) = log_b x asymptote: x = 0 domain: (0, ∞) range: (-∞, ∞) (b, 1) (1, 0) x-int y-int: none therefore, the logarithm of any base CANNOT be found for x if b = 0, b = 1, b < 0, x ≤ 0, or y = 0. VII. GRAPHING LOGARITHMIC FUNCTIONS: y = a log_b (cx-d) + e domain: (d/c, ∞) range: (-∞, ∞) VA: x = d/c ex. f(x) = log_2 (x+1) - 3 domain: (-1, ∞) range: (-∞, ∞) VA: x = -1 x-int: 7 y-int: -3 inc. or dec.: inc (-1, ∞)