Logarithmic Functions: Definitions, Properties, and Graphs

Section 5.2 Logarithmic Functions

Objective 1: Understanding the Definition of a Logarithmic Function

Logarithmic functions are the inverses of exponential functions. For any exponential function of the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$, the function is one-to-one and has an inverse function. The graph of $f(x) = a^x$ passes the horizontal line test, confirming the existence of an inverse.

• Definition: The logarithmic function with base $b$ is defined by $y = \log_b x$ if and only if $x = b^y$.

• Inverse Relationship: If $f(x) = a^x$, then $f^{-1}(x) = \log_a x$.

• Graphical Properties: The graph of $y = a^x$ contains the point $(0,1)$, and the graph of $y = \log_a x$ contains the point $(1,0)$.

• Asymptotes: The $y$-axis ($x=0$) is a vertical asymptote for $y = \log_a x$.

Example: The graph of $y = 2^x$ and its inverse $y = \log_2 x$ are reflections of each other across the line $y = x$.

Objective 2: Evaluating Logarithmic Expressions

Logarithms answer the question: "To what exponent must the base be raised to produce a given number?"

• Expression: $\log_b x$ is the exponent to which $b$ must be raised to get $x$.

• Example: $\log_2 8 = 3$ because $2^3 = 8$.

Review: Rules of Exponents and Rewriting Expressions

• Exponential expressions can be rewritten using logarithms and vice versa.

• Negative and rational exponents can be evaluated using logarithmic properties.

Additional info: See Section 5.1a for detailed rules of exponents.

Objective 3: Understanding the Properties of Logarithms

Logarithms have several important properties that simplify expressions and solve equations.

• General Properties: For $b > 0$ and $b \neq 1$:

  • $\log_b 1 = 0$

  • $\log_b b = 1$

  • $\log_b b^x = x$

  • $b^{\log_b x} = x$

• Cancellation Properties:

  • $\log_b b^x = x$

  • $b^{\log_b x} = x$

Example: $\log_3 9 = 2$ because $3^2 = 9$.

Objective 4: Using the Common and Natural Logarithms

There are two special logarithms frequently used in mathematics: the common logarithm and the natural logarithm.

• Common Logarithm: For $x > 0$, the common logarithmic function is defined by $y = \log_{10} x$ if and only if $x = 10^y$.

• Natural Logarithm: For $x > 0$, the natural logarithmic function is defined by $y = \ln x$ if and only if $x = e^y$.

Example: $\ln e = 1$ because $e^1 = e$.

Objective 5: Understanding the Characteristics of Logarithmic Functions

The graph of a logarithmic function depends on its base and has distinct characteristics.

• For $b > 1$, the logarithmic function $y = \log_b x$ has:

  • Domain: $(0, \infty)$

  • Range: $(-\infty, \infty)$

  • Vertical asymptote: $x = 0$

  • Passes through $(1, 0)$

  • Increasing on $(0, \infty)$

• For $0 < b < 1$, the graph is decreasing on $(0, \infty)$.

Example: The graph of $y = \log_2 x$ is increasing, while $y = \log_{1/2} x$ is decreasing.

Review: Using Transformations to Graph Functions

Logarithmic graphs can be shifted, stretched, or reflected using transformations.

• Horizontal shifts: $f(x) = \log_b(x - h)$ shifts the graph $h$ units to the right.

• Vertical shifts: $f(x) = \log_b x + k$ shifts the graph $k$ units up.

• Reflections: $f(x) = -\log_b x$ reflects the graph across the $x$-axis.

Example: The graph of $f(x) = \log_2(x - 1)$ is the graph of $\log_2 x$ shifted one unit to the right.

Objective 6: Sketching the Graphs of Logarithmic Functions Using Transformations

To graph $f(x) = \log_b(x - h)$, shift the graph of $\log_b x$ horizontally by $h$ units. The domain becomes $(h, \infty)$, and the vertical asymptote is $x = h$.

• Example: The graph of $f(x) = \log_2(x - 1)$ contains the points $(2, 1)$, $(3, 1.584)$, and has a vertical asymptote at $x = 1$.

Objective 7: Finding the Domain of Logarithmic Functions

The domain of a logarithmic function $g(x) = \log_b[f(x)]$ is determined by the inequality $f(x) > 0$.

• Example: For $g(x) = \log_3(x - 2)$, solve $x - 2 > 0$ to find the domain $x > 2$.

Summary Table: Properties of Logarithmic Functions

Property	Expression	Result
Product Rule	$\log_b(xy)$	$\log_b x + \log_b y$
Quotient Rule	$\log_b\left(\frac{x}{y}\right)$	$\log_b x - \log_b y$
Power Rule	$\log_b(x^r)$	$r \log_b x$
Change of Base	$\log_b x$	$\frac{\log_k x}{\log_k b}$

Additional info: The table above summarizes the main algebraic properties of logarithms used in Precalculus.