Exponential and Logarithmic Functions and Models: Precalculus Math Test – PDF Download

Question 1: Logarithm basics (2 points)

1. Write \(7^{3}=343\) in logarithmic form. (1 point)
2. Evaluate \(\log_{1/3} 9\). (1 point)

Question 2: Laws of logarithms (2 points)

1. Expand \(\ln\frac{x^{3}\sqrt{y}}{z^{2}}\). (1 point)
2. Write \(2\log a+3\log b-\log c\) as one logarithm. (1 point)

Question 3: Inverse function (2 points)

Let \(f(x)=3\ln (x-1)+2\).

1. Find \(f^{-1}(x)\). (1 point)
2. State the domain of \(f^{-1}\). (1 point)

Question 4: Graph and inverse (2 points)

The graph of an exponential function \(f\) passes through the marked points \((0,-1)\), \((1,0)\), \((2,2)\) and \((3,6)\). The dashed line is its horizontal asymptote.

1. Write an equation for \(f(x)\). (1 point)
2. Write an equation for \(f^{-1}(x)\) and give the equation of its vertical asymptote. (1 point)

Question 5: Exponential equations (2 points)

A bacteria culture has \(N=4e^{0.3t}\) thousand cells after \(t\) hours.

1. Find the time when the culture reaches 20 thousand cells. Round to two decimal places. (1 point)
2. Solve \(2^{x+1}=3^{x}\). Round to three decimal places. (1 point)

Question 6: Logarithmic equation (2 points)

Solve \(\log_{3}(x+2)+\log_{3}(x-4)=3\). Explain why any solution you reject is not valid.

Question 7: Continuous growth (2 points)

The population of a town is 12,000 and grows continuously at 2.4% per year, so \(P(t)=12000e^{0.024t}\).

1. Find the population after 10 years, to the nearest person. (1 point)
2. Find how many years it takes to reach 20,000. Round to two decimal places. (1 point)

Question 8: Logistic model (3 points)

The number of fish in a pond is \(P(t)=\frac{800}{1+7e^{-0.4t}}\), where \(t\) is in years.

1. Find the initial number of fish. (1 point)
2. Find the number of fish the pond approaches over a long time. (1 point)
3. Find when the pond holds 400 fish. Round to two decimal places. (1 point)

Question 9: Logarithmic scales (3 points)

1. An earthquake of magnitude 6.8 is compared with one of magnitude 5.6. How many times greater is the wave amplitude of the stronger one? Round to one decimal place. (1 point)
2. A solution has \([H^{+}]=2.5\times 10^{-4}\) mol/L. Find its pH to two decimal places. (1 point)
3. A siren measures 95 dB and a vacuum cleaner 80 dB. How many times greater is the siren’s intensity? Round to one decimal place. (1 point)

See every chapter on the Precalculus math page.