Unit 3: Exponential and Logarithmic Functions – Study Guide

Unit 3: Exponential and Logarithmic Functions

2.1 Exponential / Natural Log Functions

Exponential and natural logarithmic functions are fundamental in calculus, especially in business applications such as modeling growth and decay. Understanding their properties and behavior is essential for solving real-world problems.

• Exponential Function: A function of the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Natural Exponential Function: The special case where $a = e$ (Euler's number, approximately 2.718), so $f(x) = e^x$.

• Natural Logarithm: The inverse of the natural exponential function, written as $\ln(x)$, where $\ln(x) = y$ if and only if $e^y = x$.

• Properties: Exponential functions grow (or decay) rapidly, and logarithmic functions increase slowly for large $x$.

Example: $f(x) = 2^x$ is an exponential function; $g(x) = \ln(x)$ is a natural logarithmic function.

2.2/2.3 Derivatives of (Base e) and Natural Log Functions

Calculating derivatives of exponential and logarithmic functions is crucial for analyzing rates of change in business contexts, such as compound interest and population growth.

• Derivative of $e^x$: $\frac{d}{dx} e^x = e^x$

• Derivative of $a^x$: $\frac{d}{dx} a^x = a^x \ln(a)$

• Derivative of $\ln(x)$: $\frac{d}{dx} \ln(x) = \frac{1}{x}$, for $x > 0$

• Chain Rule: For composite functions, $\frac{d}{dx} e^{g(x)} = e^{g(x)} g'(x)$ and $\frac{d}{dx} \ln(g(x)) = \frac{g'(x)}{g(x)}$

Example: $\frac{d}{dx} e^{2x} = 2e^{2x}$; $\frac{d}{dx} \ln(3x) = \frac{1}{x}$

2.4 Exponential Growth Models

Exponential growth models describe processes where the rate of change is proportional to the current value, common in finance and population studies.

• General Model: $y = y_0 e^{kt}$, where $y_0$ is the initial amount, $k$ is the growth rate, and $t$ is time.

• Applications: Compound interest, population growth, and investment returns.

Example: If $y_0 = 1000$, $k = 0.05$, after 10 years: $y = 1000 e^{0.5} \approx 1649$

2.5 Exponential Decay Models

Exponential decay models are used when quantities decrease at a rate proportional to their current value, such as depreciation or radioactive decay.

• General Model: $y = y_0 e^{-kt}$, where $k > 0$ is the decay constant.

• Applications: Asset depreciation, radioactive decay, cooling processes.

Example: If $y_0 = 500$, $k = 0.1$, after 5 years: $y = 500 e^{-0.5} \approx 303$

2.6 The Derivatives of $a^x$ and $\log_a x$

This section extends differentiation to general exponential and logarithmic functions with arbitrary bases, which are important for modeling diverse business scenarios.

• Derivative of $a^x$: $\frac{d}{dx} a^x = a^x \ln(a)$

• Derivative of $\log_a x$: $\frac{d}{dx} \log_a x = \frac{1}{x \ln(a)}$

• Chain Rule Applications: For $f(x) = a^{g(x)}$, $f'(x) = a^{g(x)} \ln(a) \cdot g'(x)$

Example: $\frac{d}{dx} 5^x = 5^x \ln(5)$; $\frac{d}{dx} \log_2 x = \frac{1}{x \ln(2)}$

Unit 3 Review and Test

The review session consolidates all concepts from exponential and logarithmic functions, their derivatives, and applications in growth and decay models. The test will assess understanding and application of these topics.

• Review: Practice problems on differentiation, modeling, and interpretation of exponential/logarithmic functions.

• Test: Covers all material from Unit 3, including theoretical and applied questions.