4.2 Logarithmic Functions

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4.2.1 Learning Objectives

By the end of this section, you should be able to:

• convert between logarithmic and exponential form;
• evaluate logarithms exactly;
• graph logarithmic functions and transformations;
• determine logarithmic domains and asymptotes; and
• evaluate common and natural logarithms.

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4.2.2 Logarithmic and Exponential Forms

For \(a>0\), \(a\ne1\), and \(x>0\),

\[ \log_a x=y\quad\Longleftrightarrow\quad a^y=x. \]

A logarithm is an exponent. The logarithmic function \(f(x)=\log_a x\) is the inverse of \(a^x\).

Example 4.6 Converting Between Forms

Rewrite \(\log_3 81=4\) in exponential form and \(2^{-5}=1/32\) in logarithmic form.

Solution

\[ \boxed{3^4=81,\qquad \log_2\left(\frac1{32}\right)=-5} \]

Exercise 4.6 Rewrite \(5^3=125\) in logarithmic form.

Show answer

Answer: \(\boxed{\log_5 125=3}\)

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4.2.3 Evaluating Logarithms

To evaluate \(\log_a x\), ask: “To what exponent must \(a\) be raised to obtain \(x\)?” Important values include

\[ \log_a1=0,\qquad \log_a a=1. \]

Example 4.7 Evaluating Exact Logarithms

Evaluate \(\log_4 64\), \(\log_5(1/25)\), and \(\log_7 1\).

Solution

\[ 4^3=64,\qquad 5^{-2}=\frac1{25},\qquad 7^0=1. \]

\[ \boxed{3,\quad -2,\quad 0} \]

Exercise 4.7 Evaluate \(\log_9 3\).

Show answer

Answer: Since \(9^{1/2}=3\), \(\boxed{\log_9 3=1/2}\).

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4.2.4 Graphs and Transformations

The graph of \(y=\log_a x\) is the reflection of \(y=a^x\) across \(y=x\). Its domain is \((0,\infty)\), its range is all real numbers, its \(x\)-intercept is \((1,0)\), and \(x=0\) is a vertical asymptote.

For \(g(x)=c\log_a(x-h)+k\), transformations follow the usual rules. The vertical asymptote is \(x=h\).

Example 4.8 Graphing a Logarithmic Transformation

Describe \(g(x)=-\log_2(x+3)+1\).

Solution

The graph shifts left \(3\), reflects across the \(x\)-axis, and shifts up \(1\). The vertical asymptote is \(x=-3\), and the domain is \((-3,\infty)\).

\[ \boxed{\text{Left }3,\quad \text{reflection},\quad \text{up }1,\quad x=-3} \]

Exercise 4.8 State the domain and vertical asymptote of \(g(x)=\log_5(x-4)-2\).

Show answer

Answer: \(\boxed{\text{Domain }(4,\infty),\quad \text{asymptote }x=4}\)

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4.2.5 Common and Natural Logarithms

The common logarithm has base \(10\):

\[ \log x=\log_{10}x. \]

The natural logarithm has base \(e\):

\[ \ln x=\log_e x. \]

Because logarithms and exponentials are inverses,

\[ \ln(e^x)=x,\qquad e^{\ln x}=x\quad(x>0). \]

Example 4.9 Using Natural Logarithms

Evaluate \(\ln(e^{-3})\) and \(e^{\ln 7}\).

Solution

The inverse operations undo one another:

\[ \boxed{\ln(e^{-3})=-3,\qquad e^{\ln7}=7} \]

Exercise 4.9 Evaluate \(\log 10000\) and \(\ln1\).

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Answer: \(\boxed{\log10000=4,\quad \ln1=0}\)

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4.2.6 Domains of Logarithmic Expressions

The argument of every logarithm must be positive. To find a domain, solve

\[ \text{argument}>0. \]

Example 4.10 Finding a Logarithmic Domain

Find the domain of \(f(x)=\ln(7-2x)\).

Solution

\[ 7-2x>0\quad\Longrightarrow\quad x<\frac72. \]

\[ \boxed{(-\infty,7/2)} \]

Exercise 4.10 Find the domain of \(f(x)=\log(x^2-9)\).

Show answer

Answer: Solve \((x-3)(x+3)>0\), giving \(\boxed{(-\infty,-3)\cup(3,\infty)}\).

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4.2.7 Conceptual Takeaways

• A logarithm answers an exponent question.
• Exponential and logarithmic functions are inverses.
• Logarithmic inputs must be positive.
• A horizontal shift changes the vertical asymptote.
• Common and natural logarithms use bases \(10\) and \(e\).

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4.2.8 Skills You Should Be Able to Do

• Convert between exponential and logarithmic forms.
• Evaluate exact logarithms.
• Graph logarithmic transformations with Desmos.
• Determine logarithmic domains.
• Use common and natural logarithms.

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4.2.9 Practice Problems with Solutions

1. Rewrite \(4^3=64\) in logarithmic form.

   Show Solution

   \[ \boxed{\log_4 64=3} \]

2. Evaluate \(\log_2(1/8)\).

   Show Solution

   Since \(2^{-3}=1/8\), \(\boxed{-3}\).

3. Evaluate \(\ln(e^{5/2})\).

   Show Solution

   The functions are inverses, so \(\boxed{5/2}\).

4. State the domain and asymptote of \(f(x)=\log_3(x+5)\).

   Show Solution

   Require \(x+5>0\), so \(x>-5\). Thus,

   \[ \boxed{\text{Domain }(-5,\infty),\quad x=-5}. \]

5. Find the domain of \(f(x)=\ln(2x-1)\).

   Show Solution

   \[ 2x-1>0\quad\Longrightarrow\quad x>\frac12. \]

   \[ \boxed{(1/2,\infty)} \]

6. Use Desmos to graph \(y=\log_2x\) and \(y=2^x\). Describe their relationship.

   Show Solution

   The functions are inverses, so their graphs are reflections across \(y=x\). Points such as \((0,1)\) and \((1,0)\) exchange coordinates.

   \[ \boxed{\text{The graphs are reflections across }y=x} \]

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