Precalculus Logarithms and Their Properties Lesson (Free PDF)

Precalculus math lesson: Logarithms and Their Properties – Maths PDF

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This precalculus logarithms lesson 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.

A logarithm is an exponent: log_b x = y means b^y = x. Logarithms and exponentials are inverse functions, so log_b x is defined only for x > 0 and its graph is the reflection of b^x over y = x. The product, quotient and power rules turn multiplication into addition, division into subtraction, and powers into multiplication.

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

Key facts in this precalculus logarithms lesson

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).

What is a logarithm?

A logarithm answers the question: what exponent is needed? We define \(\log_{b}x=y\) to mean \(b^{y}=x\), where \(b>0\), \(b\ne1\) and \(x>0\). For example \(\log_{2}32=5\) because \(2^{5}=32\). The base-10 logarithm is written \(\log x\) and the base-\(e\) (natural) logarithm is written \(\ln x\).

A logarithm is an exponent.

Two values come straight from the definition: \(\log_{b}1=0\) and \(\log_{b}b=1\). Also \(b^{\log_{b}x}=x\) and \(\log_{b}b^{x}=x\), because the logarithm and the exponential undo each other.

How is a logarithm graph related to an exponential graph?

\(f(x)=b^{x}\) and \(g(x)=\log_{b}x\) are inverse functions, so their graphs are mirror images across \(y=x\). The log graph has domain \((0,\infty)\), range all real numbers, a vertical asymptote \(x=0\) and the point \((1,0)\).

The point \((3,8)\) on \(2^{x}\) matches \((8,3)\) on \(\log_{2}x\).

What are the properties of logarithms?

Rule	Formula
Product	\(\log_{b}(MN)=\log_{b}M+\log_{b}N\)
Quotient	\(\log_{b}\frac{M}{N}=\log_{b}M-\log_{b}N\)
Power	\(\log_{b}M^{p}=p\log_{b}M\)
Change of base	\(\log_{b}M=\frac{\ln M}{\ln b}\)

To expand a logarithm, apply the rules to split it into simple pieces. To condense it, work backward. Example: \(\log_{2}\frac{8x^{3}}{y}=\log_{2}8+3\log_{2}x-\log_{2}y=3+3\log_{2}x-\log_{2}y\).

\(y=\log_{3}x\) passes through \((1,0)\), \((3,1)\) and \((9,2)\): each factor of 3 adds 1.

Example (change of base): Most calculators have only \(\log\) and \(\ln\) keys. To find \(\log_{3}20\), compute \(\frac{\ln 20}{\ln 3}\approx\frac{2.9957}{1.0986}\approx 2.727\). This makes sense because \(3^{2}=9\) and \(3^{3}=27\), and 20 lies between them, so the exponent must be between 2 and 3. Estimating between two known powers is a quick way to catch a calculator mistake. You can also check your answer by raising the base to it: \(3^{2.727}\approx 20\).

Where are logarithms used?

Logarithms measure quantities that span huge ranges. Sound level in decibels is \(L=10\log\frac{I}{I_{0}}\), acidity is \(\text{pH}=-\log[\text{H}^{+}]\), and earthquake magnitude grows with the logarithm of the wave amplitude. Each step of 1 on a log scale means a factor of 10 in the quantity, so a 20 dB increase is a 100 times larger intensity.

Decibel levels: each 10 dB is ten times the intensity.

Remember that only positive numbers have logarithms. The expression \(\log(x-3)\) is defined only for \(x>3\), and a graph shifted by \(h\) has its vertical asymptote at \(x=h\).

Precalculus logarithms lesson: questions and answers

What does log base 2 of 8 mean?

It asks which power of 2 gives 8. Since 2 cubed is 8, log base 2 of 8 equals 3.

What is the difference between log and ln?

The notation log usually means base 10, and ln means base e, about 2.718. Both follow the same properties.

Why can’t you take the log of a negative number?

A positive base raised to any real power is always positive, so no exponent produces a negative number or zero. The domain of a logarithm is x > 0.

How do you evaluate a logarithm with a calculator?

Use the change of base formula. For log base 3 of 20, compute ln 20 divided by ln 3, which is about 2.727.

Does log(A + B) equal log A + log B?

No. The product rule applies to log(AB), not log(A + B). For example log(10 + 10) = log 20, not 2.

How to use this precalculus logarithms lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

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