Sample Quiz: College Algebra Chapter 6

Exponential and Logarithmic Functions — College Algebra 2e (Jay Abramson)

10 questions · 23 points · answer key included

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Which of the following correctly describes the relationship between the exponential function y = b^x and the logarithmic function y = log_b(x)? (2 PTS)
1. They are inverse functions with swapped domains and ranges
2. The logarithmic function is a horizontal shift of the exponential function
3. The exponential function is always greater than the logarithmic function
4. They are identical functions with the same domain and range
According to the product rule for logarithms, log_b(MN) can be rewritten as which of the following? (2 PTS)
1. log_b(M) - log_b(N)
2. log_b(M) + log_b(N)
3. log_b(M - N)
4. log_b(M) · log_b(N)
The equation y = 2.294e^-0.654t represents which type of exponential behavior? (2 PTS)
1. Continuous decay, because the exponent coefficient is negative
2. Continuous growth, because the base e is positive
3. Continuous growth, because the coefficient 2.294 is positive
4. Neither growth nor decay; it is a constant function
What is the domain of the logarithmic function f(x) = log(5 - 2x)? (2 PTS)
1. (0, ∞)
2. (5/2, ∞)
3. (-∞, 5/2)
4. (-∞, ∞)
Which statement correctly describes the transformation of the parent logarithmic function y = log_b(x) to obtain f(x) = log_b(x + c) + d when c > 0 and d > 0? (2 PTS)
1. Shift left c units and down d units
2. Shift right c units and down d units
3. Shift left c units and up d units
4. Shift right c units and up d units

The natural logarithm ln(x) has base e, and ln(e^x) = x for all real numbers x. (1 PTS)
Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without bound. (1 PTS)
It is reasonable to use an exponential regression model to make predictions far beyond the original observation interval. (1 PTS)

A tricycle driver in Manila invests $5,500 in a savings account earning 7.2% annual interest compounded monthly. Using the compound interest formula A(t) = P(1 + r/n)^nt, calculate the account value after 8 years. Round to the nearest peso. (5 PTS)
Solve the exponential equation 100 = 20e^2t for t. Express your answer in exact form and as a decimal rounded to four decimal places. (5 PTS)

Answer key

1.A — The exponential function y = b^x and logarithmic function y = log_b(x) are inverse functions, meaning their domains and ranges are swapped: the domain of the exponential function (-∞, ∞) becomes the range of the logarithmic function, and the range of the exponential function (0, ∞) becomes the domain of the logarithmic function.
2.B — The product rule for logarithms states that the logarithm of a product equals the sum of the logarithms: log_b(MN) = log_b(M) + log_b(N).
3.A — In the form y = A₀ e^kt, continuous decay occurs when k < 0. Here, k = -0.654, which is negative, so the function represents continuous decay.
4.C — A logarithmic function is defined only when its argument is positive. Setting 5 - 2x > 0 and solving gives x < 5/2, so the domain is (-∞, 5/2).
5.C — According to the transformations of logarithmic functions, f(x) = log_b(x + c) + d shifts the parent function left c units (when c > 0) and up d units (when d > 0).

1.True — By definition, the natural logarithm has base e, and since y = e^x and y = ln(x) are inverse functions, ln(e^x) = x for all real x.
2.True — According to the summary on fitting exponential models to data, exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down.
3.False — The textbook explicitly states that while interpolation within the original observation interval is usually reasonable, it is important to use reasoning skills to determine whether the model makes sense for inputs far beyond the original observation interval (extrapolation), and models should not be assumed to represent situations indefinitely.

1.Substituting into the compound interest formula: A(8) = 5500(1 + 0.072/12)^{12 · 8} = 5500(1 + 0.006)⁹⁶ = 5500(1.006)⁹⁶. Computing (1.006)⁹⁶ ≈ 1.7137, so A(8) ≈ 5500 × 1.7137 ≈ 9425.35. The account will be worth approximately $9,425. — The compound interest formula correctly applies the given principal, rate, compounding frequency, and time period to calculate the final account value after 8 years.
2.Divide both sides by 20: 5 = e^2t. Take the natural logarithm of both sides: ln(5) = 2t. Divide by 2: t = ln(5)/2. In decimal form, t ≈ 1.6094/2 ≈ 0.8047. — The solution correctly isolates the exponential term, applies the natural logarithm (the inverse of the exponential function with base e), and solves for t algebraically.