Logarithmic Functions Flashcards

Let f(x) = 2x + 3 and g(x) = x². What is the value of (f ∘ g)(2)?

Answer: 11

(f ∘ g)(2) = f(g(2)) = f(4) = 2(4) + 3 = 11. The inner function g is applied first.

Let f(x) = 3·2ˣ. Which expression gives the inverse function f⁻¹(x)?

Answer: log₂(x/3)

Set y = 3·2ˣ, divide by 3 to get y/3 = 2ˣ, then take log₂ of both sides: x = log₂(y/3). Swapping variables gives f⁻¹(x) = log₂(x/3).

What is the solution to log₃(x + 2) = 2?

Answer: 7

Rewriting in exponential form gives x + 2 = 3² = 9, so x = 7.

Which expression is equivalent to log(x³√y / z) for positive x, y, z?

Answer: 3 log x + ½ log y − log z

The quotient rule turns division into subtraction, the product rule turns multiplication into addition, and the power rule brings down exponents (√y = y^(1/2)).

What is the domain of f(x) = ln(5 − 2x)?

Answer: x < 5/2

The argument of a logarithm must be strictly positive, so 5 − 2x > 0, which gives x < 5/2.

What is the exact value of log₄ 32?

Answer: 5/2

Write both as powers of 2: 32 = 2⁵ and 4 = 2². Then log₄ 32 = 5/2 by the change of base formula, or because 4^(5/2) = 2⁵ = 32.

On a semi-log plot, log₁₀(y) is graphed against x, and the points form a straight line passing through (0, 1) and (2, 5). Which exponential model fits the data?

Answer: y = 10·100ˣ

The line has slope (5 − 1)/(2 − 0) = 2 and intercept 1, so log y = 2x + 1. Then y = 10^(2x+1) = 10·(10²)ˣ = 10·100ˣ.

What is the vertical asymptote of g(x) = log₂(x + 4) − 5?

Answer: x = −4

The graph of log₂ x has its asymptote at x = 0. Replacing x with x + 4 shifts it left 4 units, giving x = −4. The −5 only moves the graph vertically.

What is the solution to e^(2x) = 7?

Answer: (ln 7)/2

Take the natural log of both sides: 2x = ln 7, so x = (ln 7)/2.

Which function is the inverse of h(x) = eˣ + 1?

Answer: h⁻¹(x) = ln(x − 1)

Set y = eˣ + 1, subtract 1 to get y − 1 = eˣ, then take ln: x = ln(y − 1). Swapping variables gives h⁻¹(x) = ln(x − 1).