Precalculus Logarithms and Their Properties Test with Point Values (PDF)

Precalculus math test: Logarithms and Their Properties – Maths PDF

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This precalculus logarithms test for Precalculus covers logarithms and their properties (Common Core F-LE.A.4, F-IF.C.7e, F-BF.B.5), with a free PDF to print.

This 20-point test covers logarithm forms, properties, graphs, change of base and a pH model (F-LE.A.4, F-IF.C.7e). You have 45 minutes. Show your work.

Time and points

About this test

Grade: Precalculus
Chapter: Logarithms and Their Properties
Standards: F-LE.A.4, F-IF.C.7e, F-BF.B.5
Time: 45 minutes
Total: 20 points

Precalculus logarithms test: questions

Question 1 – Convert [2 pts]

(a) Write \(3^{4}=81\) in logarithmic form. (b) Write \(\log_{2}\frac{1}{16}=-4\) in exponential form.

Question 2 – Evaluate [2 pts]

Evaluate \(\log_{5}125\) and \(\log_{1/3}9\).

Question 3 – Domain [2 pts]

Graph of f.

Find the domain of \(f(x)=\ln(5-x)\) and the equation of its vertical asymptote.

Question 4 – Expand [2 pts]

Expand \(\log\frac{x^{2}\sqrt{y}}{z}\).

Question 5 – Read a log graph [3 pts]

Graph of a logarithm.

The graph of \(y=\log_{3}x\) is shown. (a) State the vertical asymptote. (b) State the domain and range. (c) Write the inverse function.

Question 6 – Condense [2 pts]

Write \(3\ln x-2\ln y+\ln5\) as a single logarithm.

Question 7 – Change of base [2 pts]

Evaluate \(\log_{7}50\) to three decimal places.

Question 8 – pH model [3 pts]

A pH scale.

Use \(\text{pH}=-\log[\text{H}^{+}]\). (a) Find the pH when \([\text{H}^{+}]=2.5\times10^{-5}\), to the nearest tenth. (b) Find \([\text{H}^{+}]\) when the pH is 8. (c) How many times more acidic is a solution with pH 3 than one with pH 5?

Question 9 – Use given logs [2 pts]

If \(\log_{b}2\approx0.69\) and \(\log_{b}5\approx1.61\), approximate \(\log_{b}10\) and \(\log_{b}2.5\).

Key facts behind this precalculus logarithms test

Key facts

log_b x = y if and only if b^y = x (b > 0, b not 1, x > 0).
log_b 1 = 0 and log_b b = 1.
Product rule adds logs, quotient rule subtracts, power rule multiplies.
Change of base: log_b M = ln M / ln b.
The graph of log_b x has the vertical asymptote x = 0 and passes through (1, 0).

How to use this precalculus logarithms test

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All Precalculus chapters: Precalculus math.

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