10.3 Evaluate and Graph Logarithmic Functions

Exponential and logarithmic expressions are powerful tools for modeling real-world phenomena. They allow us to describe relationships where quantities change at rates proportional to their current values, like compound interest or radioactive decay.

These functions are inverses of each other, with exponential form expressing y in terms of x, and logarithmic form expressing x in terms of y. Understanding their properties and graphs helps solve complex problems in science, finance, and engineering.

Exponential and Logarithmic Expressions

Exponential and logarithmic conversions

• Exponential form $b^x = y$ expresses a relationship where $b$ is the base, $x$ is the exponent, and $y$ is the result (2^3 = 8)
• Logarithmic form $\log_b(y) = x$ expresses the same relationship in a different way, where $b$ is the base, $y$ is the argument, and $x$ is the result ($\log_2(8) = 3$)
  • $b^x = y$ and $\log_b(y) = x$ are equivalent expressions that can be converted between each other ($2^3 = 8$ is equivalent to $\log_2(8) = 3$)
  • These expressions are inverse functions of each other
• Common logarithm $\log(x)$ is a logarithm with base 10, written as $\log_{10}(x)$ ($\log(100) = 2$ because $10^2 = 100$)
• Natural logarithm $\ln(x)$ is a logarithm with base $e$, written as $\log_e(x)$, where $e \approx 2.71828$ ($\ln(e) = 1$ because $e^1 = e$)

Evaluation of logarithmic functions

• Change of base formula $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$ allows rewriting a logarithm with base $b$ in terms of a logarithm with base $a$, where $a$ is any positive base other than 1 ($\log_5(x) = \frac{\ln(x)}{\ln(5)}$ rewrites base 5 logarithm in terms of natural logarithm)
• Product rule $\log_b(M \cdot N) = \log_b(M) + \log_b(N)$ simplifies the logarithm of a product into a sum of logarithms ($\log_2(4 \cdot 8) = \log_2(4) + \log_2(8)$)
• Quotient rule $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$ simplifies the logarithm of a quotient into a difference of logarithms ($\log_3(\frac{81}{9}) = \log_3(81) - \log_3(9)$)
• Power rule $\log_b(M^n) = n \cdot \log_b(M)$ simplifies the logarithm of a power into a product of the exponent and the logarithm of the base ($\log_2(8^3) = 3 \cdot \log_2(8)$)
• These rules are known as the properties of logarithms and are essential for simplifying and solving logarithmic equations

Graphing and Solving Logarithmic Functions

Graphs of logarithmic functions

• Parent function $f(x) = \log_b(x)$ represents the basic shape of a logarithmic function, where $b > 0$ and $b \neq 1$ ($f(x) = \log_2(x)$ is the parent function with base 2)
  • Domain $(0, \infty)$ means the function is defined for all positive real numbers ($\log_b(x)$ is undefined for $x \leq 0$)
  • Range $(-\infty, \infty)$ means the function can take on any real number value
  • Vertical asymptote $x = 0$ means the graph approaches but never touches the y-axis as $x$ approaches 0 from the right
  • Increasing if $b > 1$ means the function increases as $x$ increases (base 2 logarithm is increasing), decreasing if $0 < b < 1$ means the function decreases as $x$ increases (base 1/2 logarithm is decreasing)
• Vertical shift $f(x) = \log_b(x) + k$ moves the graph up by $k$ units ($f(x) = \log_2(x) + 1$ shifts the base 2 logarithm up by 1 unit)
• Horizontal shift $f(x) = \log_b(x - h)$ moves the graph right by $h$ units ($f(x) = \log_2(x - 3)$ shifts the base 2 logarithm right by 3 units)
• Vertical stretch/compression $f(x) = a \cdot \log_b(x)$ stretches the graph vertically by a factor of $|a|$ if $|a| > 1$ ($f(x) = 2\log_2(x)$ stretches the base 2 logarithm vertically by a factor of 2), compresses if $0 < |a| < 1$ ($f(x) = \frac{1}{2}\log_2(x)$ compresses the base 2 logarithm vertically by a factor of 1/2)
• Reflection $f(x) = -\log_b(x)$ reflects the graph over the x-axis ($f(x) = -\log_2(x)$ reflects the base 2 logarithm over the x-axis)
• A graphing calculator can be used to visualize these transformations and explore the behavior of logarithmic functions

Equations with logarithms

• Isolate the logarithm on one side of the equation to solve for the variable ($\log_3(x) + 2 = 5$ becomes $\log_3(x) = 3$)
• If the bases are the same, set the arguments equal to each other ($\log_3(x) = \log_3(27)$ means $x = 27$)
• If the bases are different, use the change of base formula to rewrite the logarithms with the same base ($\log_2(x) = \log_3(9)$ becomes $\frac{\ln(x)}{\ln(2)} = \frac{\ln(9)}{\ln(3)}$)
• Exponentiate both sides of the equation using the base of the logarithm to undo the logarithm ($\log_2(x) = 5$ becomes $2^5 = x$, so $x = 32$)

Real-world logarithmic applications

• Richter scale $M = \log_{10}(\frac{I}{I_0})$ measures the magnitude $M$ of an earthquake based on the ratio of its intensity $I$ to the threshold intensity $I_0$ (an earthquake with $I = 10,000 I_0$ has magnitude $M = \log_{10}(10,000) = 4$)
• Decibel scale $\beta = 10 \cdot \log_{10}(\frac{I}{I_0})$ measures the sound intensity level $\beta$ in decibels based on the ratio of the sound intensity $I$ to the threshold intensity $I_0$ (a sound with $I = 100 I_0$ has intensity level $\beta = 10 \cdot \log_{10}(100) = 20$ decibels)
• pH scale $\text{pH} = -\log_{10}([H^+])$ measures the acidity or basicity of a solution based on the concentration of hydrogen ions $[H^+]$ in moles per liter (a solution with $[H^+] = 10^{-7}$ mol/L has pH $= -\log_{10}(10^{-7}) = 7$, which is neutral)

Exponential Growth and Decay

• Exponential growth occurs when a quantity increases at a rate proportional to its current value, often modeled by $f(t) = A \cdot e^{kt}$ where $A$ is the initial amount, $k > 0$ is the growth rate, and $t$ is time
• Exponential decay occurs when a quantity decreases at a rate proportional to its current value, often modeled by $f(t) = A \cdot e^{-kt}$ where $A$ is the initial amount, $k > 0$ is the decay rate, and $t$ is time
• Logarithms are used to solve equations involving exponential growth or decay, such as finding the time it takes for a quantity to double or half