Precalculus Logarithms and Their Properties Worksheet Solutions (PDF)

Precalculus math worksheet solutions: Logarithms and Their Properties – Maths PDF

Cover of the book Precalculus Math Worksheets with Answers (PDF)

Download as a PDFThe Precalculus worksheets book with answersPrintable PDFSee the book ›

Sommaire

This precalculus logarithms answers 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.

Use this key to check each logarithm. Every solution shows the conversion to exponential form or the rule used, and final answers are in bold.

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

Precalculus logarithms answers: exercises 1 to 10

Exercise 1 – Exponential to log form

The base is 2, the exponent is 5 and the result is 32.

\(\log_{2}32=5\)

Exercise 2 – Log to exponential form

The base is the base; the log is the exponent.

\(3^{4}=81\) and \(5^{-2}=\frac{1}{25}\)

Exercise 3 – Evaluate a log

Ask: \(4^{?}=64\). \(4^{3}=64\).

Exercise 4 – Log of a fraction

\(2^{-3}=\frac18\).

-3

Exercise 5 – Common and natural logs

\(10^{3}=1000\). \(\ln e^{5}=5\) because \(\ln\) undoes \(e^{x}\).

\(3\) and \(5\)

Exercise 6 – A fractional answer

\(9^{1/2}=3\).

\(\frac12\)

Exercise 7 – Domain

Require \(x-3>0\).

Domain \((3,\infty)\); asymptote \(x=3\)

Exercise 8 – Expand

Quotient rule: \(\log_{2}(8x^{3})-\log_{2}y\). Product and power rules: \(\log_{2}8+3\log_{2}x-\log_{2}y\). Since \(\log_{2}8=3\):

\(3+3\log_{2}x-\log_{2}y\)

Exercise 9 – Condense

Power rule: \(\log x^{2}+\log y-\log z^{3}\). Then product and quotient rules.

\(\log\frac{x^{2}y}{z^{3}}\)

Exercise 10 – Use given values

\(\log_{b}6=0.30+0.48=0.78\). \(\log_{b}1.5=\log_{b}\frac32=0.48-0.30=0.18\). \(\log_{b}8=3(0.30)=0.90\).

\(0.78,\ 0.18,\ 0.90\)

Exercises 11 and up

Exercise 11 – Change of base

\(\frac{\ln 20}{\ln 3}=\frac{2.9957}{1.0986}\approx 2.727\).

\(2.727\)

Exercise 12 – Inverse functions

Swap the coordinates: (8, 3).

The inverse swaps coordinates: \((8,3)\). Check: \(\log_{2}8=3\).

\((8,3)\)

Exercise 13 – Transformed log graph

Asymptote x = -1; intercepts (3, 0) and (0, -2).

The graph is \(\log_{2}x\) shifted left 1 and down 2. Asymptote \(x=-1\). \(y=0\): \(\log_{2}(x+1)=2\), so \(x+1=4\), \(x=3\). \(y\)-intercept: \(\log_{2}1-2=-2\).

\(x=-1\); \((3,0)\); \((0,-2)\)

Exercise 14 – Inverse properties

Exponentials and logs undo each other, and \(\ln 1=0\).

\(7,\ 4,\ 0,\ 3x\)

Exercise 15 – Powers inside logs

\(\log_{2}16^{3}=3\cdot4=12\). \(\log0.01=\log10^{-2}=-2\). \(\left(\frac12\right)^{-3}=8\).

\(12,\ -2,\ -3\)

Exercise 16 – Decibels

The difference is \(50\) dB, so the ratio is \(10^{50/10}=10^{5}\).

\(100{,}000\) times as intense

Exercise 17 – pH

pH of about 3.5.

\(-\log(3.2\times10^{-4})\approx 3.49\). For pH 8: \([\text{H}^{+}]=10^{-8}\).

pH \(\approx 3.5\); \(10^{-8}\)

Exercise 18 – Magnitude difference

\(7-5=2\), so \(10^{2}\).

\(100\) times

Exercise 19 – Verify a property

\(12=4\cdot3\), so \(\log_{2}12=\log_{2}4+\log_{2}3=2+\log_{2}3\approx2+1.585\).

\(\approx 3.585\)

Exercise 20 – Solve for the base or argument

(a) \(x^{4}=81\), \(x=3\). (b) \(2^{-4}=\frac{1}{16}\). (c) \(5^{3}=125\).

(a) \(3\); (b) \(\frac1{16}\); (c) \(3\)

Key facts behind this precalculus logarithms answers

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 answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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

All Precalculus chapters: Precalculus math.

Voter.. post

Free printable PDF