Algebra 2 Logarithms Worksheet Solutions (PDF)

Algebra 2 math worksheet solutions: Logarithms – Maths PDF

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This algebra 2 logarithms answers for Algebra 2 covers logarithms (Common Core F-LE.A.4, F-BF.B.5, F-IF.C.7e), with a free PDF to print.

Every worksheet problem on logarithms is solved here, one step at a time. Compare your method as well as your final answer, because the steps show where applying the product rule to a sum can go wrong. Final answers appear in bold so you can find them quickly when you only need a check.

Algebra 2 · Common Core State Standards: F-LE.A.4, F-BF.B.5, F-IF.C.7e

Algebra 2 logarithms answers: exercises 1 to 10

Exercise 1 – Exponential to log

A logarithm is an exponent: log_b(x) = y means b^y = x.

The base is 2 and the exponent is 5.

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

Exercise 2 – Log to exponential

The base is 3, the answer 4 is the exponent.

\(3^{4} = 81\)

Exercise 3 – Evaluate a log

Ask: \(2^{?} = 64\). \(2^{6} = 64\).

Exercise 4 – Fractional argument

\(5^{-2} = \frac{1}{25}\).

-2

Exercise 5 – Common log

The base is 10: \(10^{3} = 1000\).

Exercise 6 – Root as an exponent

\(9^{1/2} = 3\), so the exponent is \(\frac{1}{2}\).

\(\frac{1}{2}\)

Exercise 7 – Natural log

\(\ln e^{4} = 4\) because the base is \(e\). \(\ln 1 = 0\) because \(e^{0} = 1\).

4 and 0

Exercise 8 – Solve a log equation

Rewrite: \(x = 4^{3}\).

\(x = 64\)

Exercise 9 – Read the graph

Read the y-values at x = 4 and 8.

\(f(4) = 2\) and \(f(8) = 3\), because \(2^{2} = 4\) and \(2^{3} = 8\).

The graph exists only for \(x > 0\).

f(4) = 2, f(8) = 3; domain x > 0; asymptote x = 0

Exercise 10 – Expand

Product rule: \(\log_{3}(9) + \log_{3}(x^{2})\). \(\log_{3}(9) = 2\). Power rule: \(\log_{3}(x^{2}) = 2\log_{3}(x)\).

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

Exercises 11 and up

Exercise 11 – Condense

\(2\log 5 = \log 25\). Then \(\log 25 + \log 4 = \log 100\).

\(\log 100 = 2\)

Exercise 12 – Condense a difference

Quotient rule: \(\ln\frac{12}{3} = \ln 4\).

\(\ln 4\)

Exercise 13 – Change of base

\(\log_{2}(20) = \frac{\ln 20}{\ln 2} \approx \frac{2.9957}{0.6931} \approx 4.32\).

About 4.32

Exercise 14 – Solve 2^x = 20

2^x and its inverse; 2^4.32 is about 20.

Take the log base 2 of both sides: \(x = \log_{2}(20)\).

Change of base: \(x = \frac{\ln 20}{\ln 2} \approx 4.32\).

Check: \(2^{4.32} \approx 19.97\).

\(x = \log_{2}(20) \approx 4.32\)

Exercise 15 – Coefficient in the exponent

Take the natural log: \(2x\ln 5 = \ln 100\).

\(x = \frac{\ln 100}{2\ln 5} = \frac{4.6052}{3.2189} \approx 1.43\).

x is about 1.43

Exercise 16 – Isolate first

Divide by 3: \(2^{x} = 16\). Since \(2^{4} = 16\), \(x = 4\).

\(x = 4\)

Exercise 17 – Solve a log equation, shifted

\(x + 3 = 2^{5} = 32\), so \(x = 29\). Check: \(\log_{2}(32) = 5\).

\(x = 29\)

Exercise 18 – Extraneous solution

Combine: \(\log[x(x – 3)] = 1\), so \(x(x – 3) = 10\).

\(x^{2} – 3x – 10 = 0\) gives \((x – 5)(x + 2) = 0\).

\(x = -2\) makes \(\log x\) undefined. Check \(x = 5\): \(\log 5 + \log 2 = \log 10 = 1\).

\(x = 5\)

Exercise 19 – Doubling time

Solve \(5000(1.06)^{t} = 10000\): \(1.06^{t} = 2\).

\(t = \frac{\ln 2}{\ln 1.06} \approx 11.90\). The table agrees: between 11 and 12 years.

About 11.9 years

Exercise 20 – pH scale

First: \(-\log 10^{-3} = 3\).

Second: \(-\log(2.5 \times 10^{-5}) = -(\log 2.5 – 5) = 5 – 0.398 \approx 4.60\).

pH 3 and about 4.60

Exercise 21 – Earthquakes

\(M_{1} = \log 10^{6} = 6\) and \(M_{2} = \log 10^{4} = 4\).

Intensity ratio: \(\frac{10^{6}}{10^{4}} = 100\).

Magnitudes 6 and 4; 100 times as intense

Exercise 22 – Find the error

\(\log(20) \approx 1.301\) but \(\log 10 + \log 10 = 2\). The claim is false.

The correct rule is \(\log(xy) = \log x + \log y\).

The sum rule applies to the product xy

Key facts behind this algebra 2 logarithms answers

Key facts

log_b(x) = y means b^y = x.
log_b(1) = 0 and log_b(b) = 1.
Product, quotient, and power rules: log(xy) = log x + log y, etc.
Change of base: log_b(x) = ln x / ln b.
The graph of log_b(x) is the reflection of b^x over y = x, with asymptote x = 0.

How to use this algebra 2 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.

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