📐Algebra and Trigonometry Unit 6 – Exponential & Logarithmic Functions

Key Concepts

• Exponential functions express repeated multiplication of a constant base raised to a variable exponent
• Logarithmic functions are the inverse of exponential functions, allowing us to solve for the exponent given the base and the result
• Properties of exponents simplify expressions involving products, quotients, and powers of exponential terms
  • These properties include the product rule, quotient rule, and power rule
• Properties of logarithms allow for the simplification and manipulation of logarithmic expressions
  • Key properties include the product rule, quotient rule, and power rule for logarithms
• Solving exponential equations often involves using logarithms to isolate the variable in the exponent
• Solving logarithmic equations requires using the properties of logarithms to isolate the logarithmic term and then applying the exponential function to both sides
• Real-world applications of exponential and logarithmic functions include modeling population growth, radioactive decay, and compound interest

Exponential Functions

• An exponential function is defined as $f(x) = b^x$, where $b$ is a positive constant called the base and $x$ is the exponent
• The domain of an exponential function is all real numbers, while the range is all positive real numbers
• The graph of an exponential function is always increasing if $b > 1$ and always decreasing if $0 < b < 1$
• Exponential functions have a horizontal asymptote at $y = 0$, meaning the graph approaches but never reaches the x-axis as $x$ approaches negative infinity
• The y-intercept of an exponential function is always $(0, 1)$ because $b^0 = 1$ for any base $b$
• Exponential functions exhibit a property called "exponential growth" when $b > 1$, meaning the function values increase at an increasingly rapid rate as $x$ increases
• Conversely, exponential functions exhibit "exponential decay" when $0 < b < 1$, meaning the function values decrease at a decreasing rate as $x$ increases

Properties of Exponents

• The product rule states that $b^m \cdot b^n = b^{m+n}$, where $b$ is the base and $m$ and $n$ are the exponents
• The quotient rule states that $\frac{b^m}{b^n} = b^{m-n}$, where $b$ is the base and $m$ and $n$ are the exponents
• The power rule states that $(b^m)^n = b^{mn}$, where $b$ is the base and $m$ and $n$ are the exponents
• The zero exponent rule states that $b^0 = 1$ for any base $b$ (except when $b = 0$)
• The negative exponent rule states that $b^{-n} = \frac{1}{b^n}$ for any base $b$ and exponent $n$
• These properties can be used to simplify expressions involving products, quotients, and powers of exponential terms
  • For example, $3^2 \cdot 3^4 = 3^{2+4} = 3^6 = 729$

Logarithmic Functions

• A logarithmic function is the inverse of an exponential function and is defined as $y = \log_b(x)$, where $b$ is the base and $x$ is the argument
• The logarithmic function $y = \log_b(x)$ is equivalent to the exponential equation $b^y = x$
• The domain of a logarithmic function is all positive real numbers, while the range is all real numbers
• The graph of a logarithmic function is always increasing if $b > 1$ and always decreasing if $0 < b < 1$
• Logarithmic functions have a vertical asymptote at $x = 0$, meaning the graph approaches but never reaches the y-axis as $x$ approaches 0 from the right
• The x-intercept of a logarithmic function is always $(1, 0)$ because $\log_b(1) = 0$ for any base $b$
• Common logarithms have a base of 10 and are denoted as $\log(x)$ without a subscript
• Natural logarithms have a base of $e$ (approximately 2.718) and are denoted as $\ln(x)$

Properties of Logarithms

• The product rule for logarithms states that $\log_b(M \cdot N) = \log_b(M) + \log_b(N)$
• The quotient rule for logarithms states that $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$
• The power rule for logarithms states that $\log_b(M^n) = n \cdot \log_b(M)$
• The change of base formula allows for the conversion between logarithms with different bases: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$
  • This formula is particularly useful when using a calculator that only has common or natural logarithm functions
• These properties can be used to simplify and manipulate logarithmic expressions
  • For example, $\log_2(8) + \log_2(4) = \log_2(8 \cdot 4) = \log_2(32) = 5$

Solving Exponential Equations

• To solve an exponential equation, isolate the exponential term on one side of the equation
• Apply the logarithm function to both sides of the equation, using the base of the exponential term as the base of the logarithm
  • This step "cancels out" the exponential function, leaving the exponent as a logarithmic term
• Simplify the resulting logarithmic equation using the properties of logarithms
• Solve for the variable in the simplified equation
• Check the solution by substituting it back into the original exponential equation
  • For example, to solve $2^x = 8$, apply $\log_2$ to both sides: $\log_2(2^x) = \log_2(8)$, which simplifies to $x = 3$

Solving Logarithmic Equations

• To solve a logarithmic equation, isolate the logarithmic term on one side of the equation
• Apply the exponential function to both sides of the equation, using the base of the logarithm as the base of the exponential function
  • This step "cancels out" the logarithm function, leaving the argument as an exponential term
• Simplify the resulting exponential equation using the properties of exponents
• Solve for the variable in the simplified equation
• Check the solution by substituting it back into the original logarithmic equation
  • For example, to solve $\log_3(x) = 4$, apply the exponential function with base 3 to both sides: $3^{\log_3(x)} = 3^4$, which simplifies to $x = 81$

Real-World Applications

• Exponential functions can model population growth, where the growth rate is proportional to the current population size
  • For example, the growth of bacteria in a petri dish can be modeled by $P(t) = P_0 \cdot e^{kt}$, where $P_0$ is the initial population and $k$ is the growth rate
• Radioactive decay is an example of exponential decay, where the rate of decay is proportional to the current amount of the substance
  • The half-life of a radioactive substance is the time it takes for half of the substance to decay and can be calculated using the formula $t_{1/2} = \frac{\ln(2)}{k}$
• Compound interest is calculated using an exponential function, where the interest is added to the principal at regular intervals
  • The formula for compound interest is $A = P(1 + \frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal, $r$ is the annual interest rate, $n$ 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 a logarithmic scale
  • An increase of one unit 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 a logarithmic scale
  • The pH is defined as the negative logarithm (base 10) of the hydrogen ion concentration: $pH = -\log_{10}[H^+]$
• The decibel scale, used to measure sound intensity, is a logarithmic scale where an increase of 10 decibels corresponds to a tenfold increase in sound intensity
  • The formula for decibels is $dB = 10 \log_{10}(\frac{I}{I_0})$, where $I$ is the sound intensity and $I_0$ is the reference intensity