Logarithmic functions are the inverses of exponential functions, flipping their graphs across y=x. They're defined for positive inputs and have unique properties like the product, quotient, and power rules. These functions are key to solving exponential equations and modeling real-world scenarios.

Understanding logarithms is crucial for working with exponential growth and decay. They simplify complex calculations, convert between different scales, and solve equations involving exponents. Mastering logarithms opens doors to advanced math and science applications, from finance to engineering.

Logarithmic functions as inverses

Definition and properties

• A logarithmic function is the inverse of an exponential function
  • If $y = b^x$, then $x = \log_b(y)$, where $b$ is the base of the logarithm
• The domain of a logarithmic function is the set of all positive real numbers
  • The range is the set of all real numbers
• The general form of a logarithmic function is $f(x) = \log_b(x)$
  • $b$ is a positive real number not equal to 1
  • $x$ is a positive real number
• Common logarithms have a base of 10 and are denoted as $\log(x)$ or $\log_{10}(x)$
• Natural logarithms have a base of $e$ (Euler's number, approximately 2.718) and are denoted as $\ln(x)$ or $\log_e(x)$

One-to-one property

• Logarithmic functions are one-to-one
  • For each input value, there is a unique output value, and vice versa
  • This property allows for the existence of inverse functions
• The one-to-one property ensures that logarithmic functions pass the horizontal line test
  • Any horizontal line intersects the graph of a logarithmic function at most once

Evaluating logarithmic expressions

Properties of logarithms

• The product property states that $\log_b(M \times N) = \log_b(M) + \log_b(N)$
  • $M$ and $N$ are positive real numbers
  • $b$ is the base of the logarithm
  • Example: $\log_2(4 \times 8) = \log_2(4) + \log_2(8) = 2 + 3 = 5$
• The quotient property states that $\log_b(M \div N) = \log_b(M) - \log_b(N)$
  • $M$ and $N$ are positive real numbers
  • $b$ is the base of the logarithm
  • Example: $\log_3(27 \div 9) = \log_3(27) - \log_3(9) = 3 - 2 = 1$
• The power property states that $\log_b(M^n) = n \times \log_b(M)$
  • $M$ is a positive real number
  • $n$ is any real number
  • $b$ is the base of the logarithm
  • Example: $\log_5(25^2) = 2 \times \log_5(25) = 2 \times 2 = 4$

Additional properties and formulas

• The change of base formula allows for converting between logarithms with different bases: $\log_b(x) = \log_a(x) \div \log_a(b)$
  • $a$ and $b$ are positive real numbers not equal to 1
  • $x$ is a positive real number
  • Example: $\log_2(8) = \log_10(8) \div \log_10(2) \approx 3$
• The identity property states that $\log_b(b) = 1$
  • The zero property states that $\log_b(1) = 0$
  • $b$ is a positive real number not equal to 1
• The reciprocal property states that $\log_b(1/x) = -\log_b(x)$
  • $x$ is a positive real number
  • $b$ is the base of the logarithm
  • Example: $\log_2(1/8) = -\log_2(8) = -3$

Graphing logarithmic functions

Relationship to exponential functions

• Logarithmic functions are the reflection of exponential functions across the line $y = x$
  • This relationship stems from the fact that logarithmic functions are the inverses of exponential functions
• The vertical asymptote of a logarithmic function occurs at $x = 0$
  • Logarithms are undefined for non-positive numbers
• The $x$-intercept of a logarithmic function occurs at $(1, 0)$
  • $\log_b(1) = 0$ for any base $b$

Behavior and transformations

• Logarithmic functions are increasing if the base $b$ is greater than 1
  • They are decreasing if the base $b$ is between 0 and 1
• The domain of a logarithmic function is $(0, \infty)$
  • The range is $(-\infty, \infty)$
• Transformations of logarithmic functions follow the same rules as other functions
  • $f(x) + k$ shifts the graph vertically by $k$ units
  • $f(x - h)$ shifts the graph horizontally by $h$ units
  • $-f(x)$ reflects the graph across the $x$-axis
  • $f(-x)$ reflects the graph across the $y$-axis
  • Example: The graph of $y = \log_2(x) + 1$ is shifted 1 unit up from the graph of $y = \log_2(x)$

Solving logarithmic equations

Solving techniques

• To solve logarithmic equations, isolate the logarithm on one side of the equation
  • Apply the corresponding exponential function to both sides
  • Example: To solve $\log_3(x) = 4$, apply $3^x$ to both sides: $3^{\log_3(x)} = 3^4$, which simplifies to $x = 81$
• When solving equations involving logarithms with different bases, use the change of base formula to convert all logarithms to a common base before solving
  • Example: To solve $\log_2(x) = \log_8(x - 1)$, use the change of base formula to convert $\log_8(x - 1)$ to base 2: $\log_2(x) = \log_2(x - 1) \div \log_2(8)$

Solution types and extraneous solutions

• Logarithmic equations may have multiple solutions, one solution, or no solution
  • The number of solutions depends on the domain restrictions and the values of the variables involved
• When solving logarithmic inequalities, isolate the logarithm on one side of the inequality
  • Apply the corresponding exponential function to both sides
  • Consider the direction of the inequality based on the base of the logarithm ($b > 1$ or $0 < b < 1$)
  • Example: To solve $\log_2(x) > 3$, apply $2^x$ to both sides: $x > 2^3 = 8$
• Extraneous solutions may arise when solving logarithmic equations due to the domain restrictions of logarithmic functions
  • Always check the solutions by substituting them back into the original equation to verify their validity
  • Example: When solving $\log(x - 2) + \log(x + 2) = 1$, the solution $x = 0$ is extraneous because it results in taking the logarithm of a non-positive number