Precalculus Logarithms and Their Properties Worksheet (PDF)

Precalculus math worksheet: Logarithms and Their Properties – Maths PDF

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This precalculus logarithms worksheet 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.

These 20 problems practice logarithms and their properties (Common Core F-LE.A.4, F-IF.C.7e and F-BF.B.5). You will convert between forms, evaluate logs, use the product, quotient and power rules, graph logarithms and use change of base.

Precalculus · Common Core State Standards: F-LE.A.4, F-IF.C.7e, F-BF.B.5

Precalculus logarithms worksheet: exercises 1 to 10

Exercise 1 – Exponential to log form

Write \(2^{5}=32\) in logarithmic form.

Exercise 2 – Log to exponential form

Write \(\log_{3}81=4\) and \(\log_{5}\frac{1}{25}=-2\) in exponential form.

Exercise 3 – Evaluate a log

Evaluate \(\log_{4}64\).

Exercise 4 – Log of a fraction

Values of log base 2.

Evaluate \(\log_{2}\frac18\) and compare with the table.

Exercise 5 – Common and natural logs

Evaluate \(\log 1000\) and \(\ln e^{5}\).

Exercise 6 – A fractional answer

Evaluate \(\log_{9}3\).

Exercise 7 – Domain

Find the domain of \(f(x)=\log(x-3)\) and its vertical asymptote.

Exercise 8 – Expand

Expand \(\log_{2}\frac{8x^{3}}{y}\).

Exercise 9 – Condense

Write \(2\log x+\log y-3\log z\) as a single logarithm.

Exercise 10 – Use given values

If \(\log_{b}2\approx0.30\) and \(\log_{b}3\approx0.48\), approximate \(\log_{b}6\), \(\log_{b}1.5\) and \(\log_{b}8\).

Exercises 11 and up

Exercise 11 – Change of base

Use the change of base formula to evaluate \(\log_{3}20\) to three decimal places.

Exercise 12 – Inverse functions

Graphs of y = 2^x and y = log2 x.

The point \((3,8)\) is on \(f(x)=2^{x}\). Name the matching point on \(f^{-1}(x)=\log_{2}x\) and verify it.

Exercise 13 – Transformed log graph

Graph of a shifted logarithm.

For \(y=\log_{2}(x+1)-2\), find the vertical asymptote, the \(x\)-intercept and the \(y\)-intercept.

Exercise 14 – Inverse properties

Simplify \(e^{\ln 7}\), \(10^{\log 4}\), \(\ln 1\) and \(\log_{5}5^{3x}\).

Exercise 15 – Powers inside logs

Evaluate \(\log_{2}16^{3}\), \(\log 0.01\) and \(\log_{1/2}8\).

Exercise 16 – Decibels

Common sound levels.

Sound level is \(L=10\log\frac{I}{I_{0}}\). For a rock concert, \(L=110\), and for a conversation \(L=60\). How many times more intense is the concert than the conversation?

Exercise 17 – pH

pH scale.

Acidity is \(\text{pH}=-\log[\text{H}^{+}]\). Find the pH of lemon juice with \([\text{H}^{+}]=3.2\times10^{-4}\) to the nearest tenth, and find \([\text{H}^{+}]\) when the pH is \(8\).

Exercise 18 – Magnitude difference

On a log scale, magnitude differences of 1 mean amplitudes differ by a factor of 10. How many times larger is the wave amplitude of a magnitude 7 quake than a magnitude 5 quake?

Exercise 19 – Verify a property

Show that \(\log_{2}12=2+\log_{2}3\) and approximate it using \(\log_{2}3\approx1.585\).

Exercise 20 – Solve for the base or argument

Solve: (a) \(\log_{x}81=4\); (b) \(\log_{2}x=-4\); (c) \(\log_{5}125=x\).

Key facts behind this precalculus logarithms worksheet

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 worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the precalculus logarithms worksheet as a PDF with the button below and keep it with your notes.

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