1.3 Exponential and Logarithmic Functions

Exponential and logarithmic functions are powerful tools in calculus, modeling real-world phenomena like population growth and radioactive decay. These functions are inverses of each other, with exponential functions growing or decaying rapidly and logarithmic functions increasing more slowly.

Understanding these functions is crucial for solving complex equations and analyzing data in fields like finance, science, and engineering. They provide a foundation for more advanced calculus concepts and are essential for modeling many natural and economic processes.

Exponential Functions

Definition and Characteristics

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• Exponential function takes the form $f(x) = b^x$, where $b$ is a positive real number not equal to 1 called the base and $x$ is any real number
• Natural exponential function is a special case where the base is the mathematical constant $e$ (approximately 2.71828), expressed as $f(x) = [e^x](https://www.fiveableKeyTerm:e^x)$
• Exponential functions are always positive, never zero or negative, as the base is raised to a power
• Exponential functions have a horizontal asymptote at $y = 0$ when the base is between 0 and 1, and no horizontal asymptote when the base is greater than 1

Growth and Decay Models

• Exponential growth occurs when the rate of change is proportional to the current value, resulting in the function increasing at an increasingly rapid rate ($b > 1$)
• Exponential decay happens when the rate of change is proportional to the current value, causing the function to decrease at a decreasing rate ($0 < b < 1$)
• Real-world applications of exponential growth include population growth, compound interest, and viral spread (COVID-19 pandemic)
• Exponential decay models can represent radioactive decay, cooling of objects, and drug elimination from the body (half-life of medication)

Solving Exponential Equations

• Exponential equations involve the variable in the exponent, such as $2^x = 8$
• To solve exponential equations with the same base on both sides, set the exponents equal and solve for the variable ($2^x = 2^3$ implies $x = 3$)
• When the bases are different, use logarithms to rewrite the equation and solve for the variable ($2^x = 8$ can be rewritten as $\log_2 8 = x$, giving $x = 3$)
• Applications of exponential equations include determining doubling time, half-life, and carbon dating (estimating the age of organic materials)

Logarithmic Functions

Definition and Properties

• Logarithm is the inverse operation of exponentiation, denoted as $\log_b x = y$ if and only if $b^y = x$, where $b$ is the base, $x$ is the argument, and $y$ is the logarithm
• Natural logarithm ($\ln$) is the logarithm with base $e$, the mathematical constant approximately equal to 2.71828
• Logarithmic properties include product rule ($\log_b (MN) = \log_b M + \log_b N$), quotient rule ($\log_b (M/N) = \log_b M - \log_b N$), and power rule ($\log_b (M^n) = n \log_b M$)
• Logarithms can be used to simplify calculations involving large numbers or powers (pH scale, Richter scale for earthquake magnitudes)

Solving Logarithmic Equations

• Logarithmic equations have the variable inside a logarithm, such as $\log_2 x = 3$
• To solve logarithmic equations, rewrite the equation in exponential form and solve for the variable ($\log_2 x = 3$ becomes $2^3 = x$, giving $x = 8$)
• When multiple logarithms are present, use the properties of logarithms to combine or separate terms before solving ($\log_3 x + \log_3 (x-1) = 2$ can be rewritten as $\log_3 (x(x-1)) = 2$)
• Applications of logarithmic equations include determining the time required for an investment to reach a specific value or the intensity of sound (decibel scale)

Logarithmic and Exponential Relationships

Change of Base Formula

• Change of base formula allows converting logarithms from one base to another: $\log_b x = \frac{\log_a x}{\log_a b}$, where $a$ is any positive real number not equal to 1
• Common use is to rewrite logarithms with base 10 or base $e$ in terms of natural logarithms or common logarithms ($\log_2 x = \frac{\ln x}{\ln 2}$)
• The change of base formula is derived from the properties of logarithms and exponential functions
• Applying the change of base formula simplifies calculations when using logarithmic or exponential models (Fibonacci sequence, golden ratio)