Logarithms and Their Properties: Precalculus Study Notes

Logarithms

Definition and Basic Properties

Logarithms are mathematical functions that answer the question: "To what exponent must a base be raised to produce a given number?" For all real numbers y, and all positive numbers a and x (where a ≠ 1):

• Definition: $y = \log_a x$ if and only if $x = a^y$

• Common Logarithms: Base 10, written as $\log x$

• Natural Logarithms: Base $e$, written as $\ln x$

Calculator Usage

Most calculators can evaluate two types of logarithms:

Type	Notation	Calculator Key
Common Logs	$\log_{10} x$	log x
Natural Logs	$\log_e x$	ln x

Examples: Evaluating Logarithms

• $\log_{10} 2.5$

• $\log_{10} \left(\frac{1}{3}\right)$

• $\ln 0.3$

• $\ln(-1)$ (Note: The natural logarithm is undefined for negative arguments.)

Properties of Logarithms

Logarithms have several important properties, which also apply to natural logarithms:

Property	Explanation
$\log_a 1 = 0$	Because $a^0 = 1$
$\log_a a = 1$	Because $a^1 = a$
$\log_a a^x = x$	Because $a^x = a^x$

Graphs of Logarithmic Functions

Key Features of Logarithmic Graphs

The graph of a logarithmic function $y = \log_a x$ has several distinctive features:

• Key Points: The points $(1, 0)$, $(a, 1)$, and $\left(\frac{1}{a}, -1\right)$ are on the graph.

• Increasing/Decreasing: If $a > 1$, the function is increasing; if $0 < a < 1$, the function is decreasing.

• Vertical Asymptote: The y-axis ($x = 0$) is a vertical asymptote.

• Domain: $(0, \infty)$

• Range: $(-\infty, \infty)$

Examples of Logarithmic Functions

• $y = \log x$

• $y = \log_2(x - 1) + 3$

These examples illustrate transformations such as horizontal shifts and vertical translations.

Solving Logarithmic and Exponential Equations

Logarithmic Equations

To solve equations involving logarithms, rewrite the equation in exponential form or use properties of logarithms.

• Example: Solve $\log_3(4x - 7) = 2$ Solution: Rewrite in exponential form: $4x - 7 = 3^2 = 9$ $4x = 16$ $x = 4$

Exponential Equations

Exponential equations can often be solved by taking the logarithm of both sides.

• Example: Solve $e^{2x} = 5$ Solution: Take the natural logarithm of both sides: $2x = \ln 5$ $x = \frac{\ln 5}{2}$

Additional info: The notes do not cover logarithm laws such as product, quotient, and power rules, but these are essential for more advanced problems.

----------------------------------------