📏Honors Pre-Calculus Unit 4 – Exponential and Logarithmic Functions

Key Concepts

• Exponential functions involve a constant base raised to a variable power and have the form $f(x) = b^x$, where $b$ is a positive real number not equal to 1
• Logarithmic functions are the inverse of exponential functions and have the form $f(x) = \log_b(x)$, where $b$ is the base and $x$ is a positive real number
• Properties of exponents simplify expressions involving exponents and include the product rule, quotient rule, and power rule
• Exponential growth occurs when a quantity increases by a constant percent over equal time intervals, while exponential decay occurs when a quantity decreases by a constant percent over equal time intervals
• Logarithms are exponents and can be used to solve equations involving exponential expressions
• Properties of logarithms, such as the product rule, quotient rule, and power rule, simplify logarithmic expressions and solve equations
• Exponential and logarithmic equations can be solved using the properties of exponents, properties of logarithms, and algebraic techniques
• Real-world applications of exponential and logarithmic functions include population growth, radioactive decay, compound interest, and the Richter scale for measuring earthquakes

Exponential Functions

• Exponential functions have the form $f(x) = b^x$, where $b$ is a positive real number not equal to 1 and is called the base
• The domain of an exponential function is all real numbers, and the range is all positive real numbers
• When the base $b$ is greater than 1, the exponential function is increasing, and when $0 < b < 1$, the function is decreasing
• The graph of an exponential function is always above the x-axis and never touches it, as the range is all positive real numbers
• The y-intercept of an exponential function is always $(0, 1)$ because $b^0 = 1$ for any base $b$
• Exponential functions have a horizontal asymptote at $y = 0$, meaning the graph gets closer to the x-axis as $x$ approaches negative infinity
• The general form of an exponential function is $f(x) = ab^x$, where $a$ is the vertical stretch factor and $b$ is the base
  • If $|a| > 1$, the graph is stretched vertically by a factor of $a$
  • If $0 < |a| < 1$, the graph is compressed vertically by a factor of $a$

Properties of Exponents

• The product rule states that when multiplying two powers with the same base, add the exponents: $b^m \cdot b^n = b^{m+n}$
• The quotient rule states that when dividing two powers with the same base, subtract the exponents: $\frac{b^m}{b^n} = b^{m-n}$
• The power rule states that when raising a power to a power, multiply the exponents: $(b^m)^n = b^{mn}$
• The zero exponent rule states that any base raised to the power of 0 equals 1: $b^0 = 1$ (except when $b = 0$)
• The negative exponent rule states that a base raised to a negative exponent is equal to its reciprocal raised to the positive exponent: $b^{-n} = \frac{1}{b^n}$
• The fractional exponent rule states that a base raised to a fractional exponent is equivalent to taking the root of the base and then raising it to the power of the numerator: $b^{\frac{m}{n}} = \sqrt[n]{b^m} = (\sqrt[n]{b})^m$
• When multiplying powers with the same exponent but different bases, multiply the bases and keep the exponent: $a^n \cdot b^n = (ab)^n$

Exponential Growth and Decay

• Exponential growth occurs when a quantity increases by a constant percent over equal time intervals, and the growth factor is greater than 1
  • The exponential growth function is $A(t) = A_0(1 + r)^t$, where $A_0$ is the initial amount, $r$ is the growth rate (as a decimal), and $t$ is the time
• Exponential decay occurs when a quantity decreases by a constant percent over equal time intervals, and the decay factor is between 0 and 1
  • The exponential decay function is $A(t) = A_0(1 - r)^t$, where $A_0$ is the initial amount, $r$ is the decay rate (as a decimal), and $t$ is the time
• The half-life of a substance is the time it takes for half of the substance to decay, while the doubling time is the time it takes for a quantity to double in size
• Continuous growth or decay is modeled using the exponential function $A(t) = A_0e^{kt}$, where $e$ is the mathematical constant approximately equal to 2.71828, and $k$ is the continuous growth or decay rate
• Real-world examples of exponential growth include population growth and compound interest, while examples of exponential decay include radioactive decay and medication concentration in the body

Introduction to Logarithms

• Logarithms are exponents and are used to solve equations involving exponential expressions
• The logarithm of a number $x$ with base $b$ is the exponent to which $b$ must be raised to get $x$, denoted as $\log_b(x) = y$, where $b^y = x$
• The base $b$ of a logarithm must be a positive real number not equal to 1, and the argument $x$ must be a positive real number
• Common logarithms have a base of 10 and are denoted as $\log(x)$, while natural logarithms have a base of $e$ (approximately 2.71828) and are denoted as $\ln(x)$
• Logarithms can be used to convert between exponential and logarithmic forms of an equation, such as $b^y = x$ and $\log_b(x) = y$
• The logarithm of 1 with any base is always 0 because any base raised to the power of 0 equals 1: $\log_b(1) = 0$
• The logarithm of the base with the same base is always 1 because any base raised to the power of 1 equals itself: $\log_b(b) = 1$

Logarithmic Functions

• Logarithmic functions are the inverse of exponential functions and have the form $f(x) = \log_b(x)$, where $b$ is the base and $x$ is a positive real number
• The domain of a logarithmic function is all positive real numbers, and the range is all real numbers
• The graph of a logarithmic function is the reflection of the corresponding exponential function over the line $y = x$
• Logarithmic functions have a vertical asymptote at $x = 0$ because the argument of a logarithm cannot be zero or negative
• The x-intercept of a logarithmic function is always $(1, 0)$ because $\log_b(1) = 0$ for any base $b$
• The general form of a logarithmic function is $f(x) = a \log_b(x)$, where $a$ is the vertical stretch factor and $b$ is the base
  • If $|a| > 1$, the graph is stretched vertically by a factor of $a$
  • If $0 < |a| < 1$, the graph is compressed vertically by a factor of $a$
• Logarithmic functions can be transformed horizontally and vertically using the same techniques as other functions, such as $f(x) = \log_b(x - h) + k$, where $h$ is the horizontal shift and $k$ is the vertical shift

Properties of Logarithms

• The product rule states that the logarithm of a product is equal to the sum of the logarithms of the factors: $\log_b(MN) = \log_b(M) + \log_b(N)$
• The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator: $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$
• The power rule states that the logarithm of a power is equal to the exponent times the logarithm of the base: $\log_b(M^n) = n \log_b(M)$
• The change of base formula allows for converting logarithms between different bases: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$, where $a$ is any positive real number not equal to 1
• The logarithm of a reciprocal is the negative of the logarithm of the original number: $\log_b(\frac{1}{M}) = -\log_b(M)$
• The logarithm of a root is equal to the logarithm of the radicand divided by the index of the root: $\log_b(\sqrt[n]{M}) = \frac{1}{n} \log_b(M)$
• These properties can be used to simplify logarithmic expressions, solve equations, and convert between logarithms with different bases

Solving Exponential and Logarithmic Equations

• To solve exponential equations, isolate the exponential expression on one side of the equation and take the logarithm of both sides
  • For example, to solve $2^x = 8$, take the logarithm (base 2) of both sides: $\log_2(2^x) = \log_2(8)$, which simplifies to $x = 3$
• To solve logarithmic equations, isolate the logarithmic expression on one side of the equation and rewrite it in exponential form
  • For example, to solve $\log_3(x) = 4$, rewrite it as an exponential equation: $3^4 = x$, which simplifies to $x = 81$
• When solving equations involving both exponential and logarithmic expressions, use the properties of exponents and logarithms to simplify the equation before solving
• Be aware of the domain restrictions when solving exponential and logarithmic equations, as the argument of a logarithm must be positive, and the base must be positive and not equal to 1
• Extraneous solutions may arise when solving exponential and logarithmic equations, so it is important to check the solutions in the original equation to verify their validity
• When solving equations with multiple logarithms, it may be necessary to use the change of base formula to convert all logarithms to a common base before applying the properties of logarithms
• Graphing can be a useful tool for estimating solutions to exponential and logarithmic equations, especially when exact solutions are difficult to find algebraically

Real-World Applications

• Exponential functions can model population growth, where the growth rate is proportional to the current population size
  • The exponential growth model is $P(t) = P_0e^{rt}$, where $P_0$ is the initial population, $r$ is the growth rate, and $t$ is the time
• Radioactive decay is modeled by exponential functions, where the decay rate is proportional to the current amount of the substance
  • The half-life formula is $A(t) = A_0(\frac{1}{2})^{\frac{t}{t_{1/2}}}$, where $A_0$ is the initial amount, $t$ is the time, and $t_{1/2}$ is the half-life
• Compound interest is calculated using exponential functions, where the interest is added to the principal at regular intervals
  • The compound interest formula is $A(t) = P(1 + \frac{r}{n})^{nt}$, where $P$ is the principal, $r$ is the annual interest rate (as a decimal), $n$ is the number of compounding periods per year, and $t$ is the time (in years)
• The Richter scale, used to measure the magnitude of earthquakes, is based on a logarithmic scale
  • An increase of 1 on the Richter scale corresponds to a tenfold increase in the amplitude of the seismic waves and a 32-fold increase in the energy released
• The pH scale, used to measure the acidity or basicity of a solution, is also based on a logarithmic scale
  • The pH of a solution is defined as the negative logarithm (base 10) of the hydrogen ion concentration: $pH = -\log_{10}[H^+]$
• Logarithmic functions can model the relationship between the perceived loudness of a sound (in decibels) and its intensity
  • The decibel scale is defined as $L = 10 \log_{10}(\frac{I}{I_0})$, where $L$ is the loudness in decibels, $I$ is the intensity of the sound, and $I_0$ is the reference intensity (typically the threshold of human hearing)
• Exponential and logarithmic functions have numerous applications in fields such as physics, chemistry, biology, economics, and computer science, making them essential tools for modeling and problem-solving in various real-world contexts