Logarithmic Functions

Definition and Inverse Relationship

The logarithmic function with base b is defined as the inverse of the exponential function with the same base. If the exponential function is one-to-one (which it is for all valid bases), then its inverse exists and is the logarithmic function.

• Domain of the logarithmic function:

• Range of the logarithmic function:

• Inverse relationship: If , then

Logarithmic and Exponential Forms

The logarithmic function with base b is written as , where and , and (since the domain is ). The equivalence between logarithmic and exponential forms is:

• is equivalent to

Examples:

• is equivalent to

• means

• means

Conversely, converting from exponential to logarithmic form:

• means

• means

• means (also written as )

Evaluating Logarithms

To evaluate logarithms, rewrite the equation in exponential form and solve for the unknown.

• Example a:

  • Let . Then . Since , .

• Example b:

  • Let . Then . Since , .

• Example c:

  • Let . Then . Since , .

• Example d:

  • Let . Then . Since , .

Special Cases:

• Natural logarithm: When the base is , is written as .

• Common logarithm: When the base is $10\log_{10} x\log x$.

Properties of Logarithms

• Identity property: because

• Zero property: because

Inverse Properties of Logarithms

These properties show that logarithms and exponentials are inverse operations and "undo" each other.

• For example, and

• and

Graphing Logarithmic Functions

General Characteristics

The graph of is smooth and continuous, with no sharp corners or gaps. It is the reflection of the exponential function over the line .

• Domain:

• Range:

• x-intercept: (since )

• No y-intercept

• Vertical asymptote:

Case 1:

• The function is increasing for .

• The graph passes through and approaches the y-axis as a vertical asymptote.

Case 2:

• The function is decreasing for .

• The graph passes through and approaches the y-axis as a vertical asymptote.

Finding Domains of Logarithmic Functions

• For , set . Domain:

• For , set . Domain:

• For , set . Domain:

• Logarithms are undefined for negative arguments (e.g., is not defined).

Transformations of Logarithmic Functions

Vertical Transformations (Changes in y)

• shifts the graph up by units

• shifts the graph down by units

• reflects the graph about the x-axis

• produces a vertical stretch if and a vertical shrink if

Horizontal Transformations (Changes in x)

• shifts the graph left by units

• shifts the graph right by units

• reflects the graph about the y-axis

• produces a horizontal shrink if and a horizontal stretch if

Summary Table: Key Properties of Logarithmic Functions