Exponential and Logarithmic Functions: Properties, Graphs, and Domains

Exponential and Logarithmic Functions

Introduction to Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. They are fundamental in modeling growth and decay processes in mathematics and science.

• General Form: $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Examples: $h(x) = 3^{x^2 + 1}$, $f(x) = 4^x$, $g(x) = 2^x$.

• Key Properties:

  • Domain: $(-\infty, \infty)$

  • Range: $(0, \infty)$

  • Horizontal asymptote: $y = 0$

Logarithmic Notation and Definition

Logarithms are the inverse operations of exponentials. The logarithm of a number is the exponent to which the base must be raised to produce that number.

• Definition: $\log_b n = x$ means $b^x = n$.

• Base: The number $b$ in $\log_b n$.

• Exponent: The value $x$ in $b^x = n$.

• Value: The result $n$.

Exponential Form	Logarithmic Form
$a^n = b$	$\log_a b = n$

Converting Between Exponential and Logarithmic Forms

It is important to be able to rewrite equations between exponential and logarithmic forms.

• Exponential to Logarithmic: $a^x = b \rightarrow \log_a b = x$

• Logarithmic to Exponential: $\log_a b = x \rightarrow a^x = b$

• Example: $2 = \log_x y \rightarrow x^2 = y$

Evaluating Logarithmic Expressions

Basic Evaluation

Logarithmic expressions can be evaluated using properties of exponents and logarithms.

• Examples:

  • $\log_2 16 = 4$ because $2^4 = 16$

  • $\log_3 1 = 0$ because $3^0 = 1$

  • $\log_7 7 = 1$ because $7^1 = 7$

  • $\log_2 5$ is irrational; it can be left in logarithmic form or approximated.

  • $\log_2 \sqrt{2} = 0.5$ because $2^{0.5} = \sqrt{2}$

Inverse Properties of Logarithms

Logarithms and exponentials are inverse functions. Their properties allow simplification of expressions.

• Inverse Properties:

  • $\log_a a^x = x$

  • $a^{\log_a x} = x$

  • If $\log_a x = \log_a y$, then $x = y$ (one-to-one property)

• Example: $\log_4 4^5 = 5$

• Example: $6^{\log_6 9} = 9$

Graphs of Logarithmic Functions

Basic Graphs and Properties

Logarithmic functions are the inverses of exponential functions. Their graphs have distinct characteristics.

• General Form: $f(x) = \log_b x$

• Domain: $(0, \infty)$

• Range: $(-\infty, \infty)$

• Vertical asymptote: $x = 0$

• Key points: $(1, 0)$, $(b, 1)$

Common Logarithms and Natural Logarithms

Definitions and Notation

There are two frequently used logarithms: common logarithms (base 10) and natural logarithms (base $e$).

Type	Notation	Base
Common Logarithm	$\log x$	10
Natural Logarithm	$\ln x$	$e \approx 2.718$

• Example: $\log_{10} 100 = 2$

• Example: $\ln e = 1$

Domain of Logarithmic Functions

Finding the Domain

The domain of a logarithmic function $f(x) = \log_b (g(x))$ consists of all $x$ for which $g(x) > 0$.

• Example: $f(x) = \log_2 (x + 3)$ has domain $x + 3 > 0 \Rightarrow x > -3$

• Example: $g(x) = \log_2 (4 - x)$ has domain $4 - x > 0 \Rightarrow x < 4$

• Example: $f(x) = \ln(2 - x)$ has domain $2 - x > 0 \Rightarrow x < 2$

• Example: $g(x) = \ln((x - 3)^2)$ has domain $(x - 3)^2 > 0 \Rightarrow x \neq 3$

Properties and Simplification of Logarithmic Expressions

Key Properties

• $\log_b 1 = 0$

• $\log_b b = 1$

• $\log_b b^x = x$

• $b^{\log_b x} = x$

• $\ln 1 = 0$

• $\ln e = 1$

• $\ln e^x = x$

• $e^{\ln x} = x$

Examples of Simplification

• $\log 10^7 = 7$

• $\ln \frac{1}{e^2} = -2$

• $\log 100 = 2$

• $e^{\ln 300} = 300$

• $\ln 1 = 0$

• $\ln e = 1$

• $\ln e^x = x$

Summary Table: Logarithmic Properties

Property	Equation
Logarithm of 1	$\log_b 1 = 0$
Logarithm of base	$\log_b b = 1$
Inverse property	$\log_b b^x = x$
Exponential inverse	$b^{\log_b x} = x$
Natural log of 1	$\ln 1 = 0$
Natural log of $e$	$\ln e = 1$
Natural log inverse	$\ln e^x = x$
Exponential inverse	$e^{\ln x} = x$

Additional info:

• All logarithmic functions require their argument to be positive.

• Logarithms are undefined for zero and negative numbers in the real number system.

• Common logarithms ($\log x$) are used in scientific calculations, while natural logarithms ($\ln x$) are prevalent in calculus and higher mathematics.