University:
High School
Course:
High School Mathematics
Academic year:

2022
Views:

331

Pages:

5
Author:

Wildmotivbu7o

Equations and their corresponding graphs/solutions: (e^x = 2) Graph: A Solution: x ≈ 0.693 (ln x = -1) Graph: F Solution: x = 1/e (2^x = 3^{-x}) Graph: B Solution: x = 0 (log_4 x = 1) Graph: E Solution: x = 4 (log_5 x = \frac{1}{2}) Graph: D Solution: x = √5 (4^x = 2) Graph: C Solution: x = 1/2 Vocabulary Terms: Exponential Equations: when the variable is in the "exponent" spot Logarithmic Equations: y = logb x Core Concepts: Property of Equality for Exponential Equations: Algebra: If b is a positive real number other than 1, then b^x = b^y if and only if x = y. Example: If 3^x = 3^5, then x = 5. When Solving: Focus on Exponents So as long as the bases are the same then just focus on the exponents. (we really don't care what the base is) Property of Equality for Logarithmic Equations: Algebra: If b, x, and y are positive real numbers with b ≠ 1, then logb x = logb y if and only if x = y. Example: If log2 x = log2 7, then x = 7. When Solving: Same as the Exponential equation. If the log bases are the same then just focus on the expression Instructions: Solve the equation. Equations and Solutions: (5^{2x+4} = 5^{5x-8}) "Same bases ... So ignore" 2x + 4 = 5x - 8 -2x -2x 4 = 3x - 8 +8 +8 12 = 3x 12/3 = 3x/3 4 = x (4^{2x-1} = 8^{x+2}) (2²)^(2x-1) = (2³)^(x+2) "now the bases are the same" 2(2x-1) = 3(x+2) 4x-2 = 3x+6 -3x -3x x-2 = 6 x = 8 ((3^{x+3}) = (5)) x+3 = log₃5 -3 -3 x = log₃5 - 3 ((\frac{1}{5})^{3x-2} = \sqrt{25^{x}}) "rewrite" (5⁻¹)^(3x-2) = (25ˣ)^(1/2) "Same bases" -1(3x-2) = x(1/2) -3x+2 = x +3x +3x 2 = 4x 2/4 = x 1/2 = x (12e^{1-2x} = 500) e^(1-2x) = 125/3 1-2x = ln(125/3) -2x = ln(125/3) - 1 x = -1/2 * [ln(125/3) - 1] x = 1 - 1/2 * ln(125/3) -14 + 3eˣ = 11 3eˣ = 25 eˣ = 25/3 x = ln(25/3)