Exponential and Logarithmic Functions: Definitions, Properties, and Applications

Exponential Functions

Definition and Exponential Behavior

Exponential functions are widely used in mathematics, science, and engineering to model growth and decay processes. An exponential function with base a is defined as a function of the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$. These functions are characterized by the property that the rate of change of the function is proportional to its current value.

• Key Property: If a quantity increases by a fixed percentage over equal time intervals, it can be modeled by an exponential function.

• Doubling and Halving: Exponential growth and decay can be described in terms of doubling time (for growth) or half-life (for decay).

• General Form: $f(x) = a^x$

Applications: Investment Growth Example

Exponential functions are commonly used to model compound interest in finance. The amount $A$ in an account after $t$ years, with principal $P$ and annual interest rate $r$, is given by:

$A = P(1 + r)^t$

• Example: If $100$is invested at $5\%$ annual interest, the amount after $t$ years is $A = 100(1.05)^t$.

Table: Investment Account Growth

The following table illustrates the growth of an investment account over several years:

Year	Amount (dollars)	Yearly increase
2022	100	—
2023	100(1.05) = 105.00	5.00
2024	100(1.05)^2 = 110.25	5.25
2025	100(1.05)^3 = 115.76	5.51
2026	100(1.05)^4 = 121.55	5.79

Properties of Exponents

Exponential expressions follow specific algebraic rules:

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: $(a^m)^n = a^{mn}$

• Zero Exponent: $a^0 = 1$

• Negative Exponent: $a^{-n} = \frac{1}{a^n}$

The Natural Exponential Function

The most important exponential function is the natural exponential function, where the base is the special number $e \approx 2.71828$. The function $f(x) = e^x$ is used extensively in calculus and mathematical modeling.

• Continuous Growth/Decay: Many natural processes are best modeled using $e^x$ due to its unique calculus properties.

Exponential Growth and Decay

Exponential functions model both growth and decay:

• Exponential Growth: $y = y_0 e^{kt}$, where $k > 0$

• Exponential Decay: $y = y_0 e^{-kt}$, where $k > 0$

• Half-life: The time required for a quantity to decrease to half its initial value.

Inverse Functions and Logarithms

One-to-One and Inverse Functions

A function is one-to-one if each output is produced by exactly one input. The inverse function $f^{-1}$ reverses the effect of $f$, so that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

• Horizontal Line Test: A function is one-to-one if every horizontal line intersects its graph at most once.

Logarithmic Functions

The logarithmic function with base $a$ is the inverse of the exponential function $a^x$. It is defined as:

$y = \log_a x \iff x = a^y$

• Natural Logarithm: The logarithm with base $e$ is called the natural logarithm and is written as $\ln x$.

• Common Logarithm: The logarithm with base $10$ is called the common logarithm and is written as $\log x$.

Properties of Logarithms

Logarithms satisfy several important properties:

• Product Rule: $\log_a(xy) = \log_a x + \log_a y$

• Quotient Rule: $\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$

• Power Rule: $\log_a(x^r) = r \log_a x$

• Change-of-Base Formula: $\log_a x = \frac{\log_b x}{\log_b a}$

Applications of Logarithms

Logarithms are used to solve equations involving exponential growth and decay, such as finding the time required for an investment to reach a certain value or the half-life of a radioactive substance.

• Example: If $5000$is invested at $3.75\%$ interest compounded annually, the time to reach $5200$is found by solving $5200 = 5000(1.0375)^t$ for $t$ using logarithms.

• Half-life Formula: $t = \frac{\ln(1/2)}{-k}$, where $k$ is the decay constant.

Inverse Trigonometric Functions (Mentioned)

Inverse trigonometric functions are introduced as the inverses of the basic trigonometric functions, but their detailed study is reserved for later chapters.

Additional info: The notes above include all main concepts, definitions, properties, and applications of exponential and logarithmic functions as presented in the provided materials. Tables and graphs are included where they directly support the explanation of the mathematical concepts.