Logarithmic Properties (6.5)

Domain of the Function

• Understanding the domain of a function is crucial as it defines the set of input values (x-values) for which the function is defined.

• Discussing the results of the domain typically involves analyzing the function for restrictions such as division by zero or taking square roots of negative numbers.

• Practice Problem: Finding Domain

  • Problem: Find the domain of $f(x) = \frac{7}{x - 5}$.

  • Step 1: Identify the restriction. The denominator cannot be zero: $x - 5 \neq 0$.

  • Step 2: Solve for x: $x \neq 5$.

  • Solution: The domain is all real numbers except 5, expressed as $(-\infty, 5) \cup (5, \infty)$.

Finding x-intercepts of the Function

• To find the x-intercepts algebraically:

  • Set the function equal to zero and solve for x.

  • Use properties of exponents to simplify where necessary.

  • It is vital to show each step clearly to track the solution process.

  • After finding the x-intercepts, verify the results by graphing the function to see if it crosses the x-axis at the calculated points.

• Practice Problem: Finding x-intercepts

  • Problem: Find the x-intercept of $f(x) = 2^x - 16$.

  • Step 1: Set $f(x) = 0$: $0 = 2^x - 16$.

  • Step 2: Isolate the exponential part: $2^x = 16$.

  • Step 3: Rewrite 16 as a power of 2: $2^x = 2^4$.

  • Solution: $x = 4$. The x-intercept is $(4, 0)$.

6.5 Logarithmic Properties

Learning Objectives

• In this section, students will learn how to:

  • Use the product rule for logarithms.

  • Use the quotient rule for logarithms.

  • Use the power rule for logarithms.

  • Expand logarithmic expressions.

  • Condense logarithmic expressions.

  • Use the change-of-base formula for logarithms.

pH Scale and Logarithms

• Definition of pH: pH is a measure of the acidity or alkalinity of a substance measured on a scale from 0 to 14.

  • Substances with a pH < 7 are acidic.

  • Substances with a pH > 7 are basic.

  • The human body requires a pH close to 7.35 for enzyme function.

• Common pH Levels:

  • Battery acid: 0.8

  • Stomach acid: 2.7

  • Orange juice: 3.3

  • Pure water: 7 (at 25°C)

  • Human blood: 7.35

  • Fresh coconut: 7.8

  • Sodium hydroxide (lye): 14

• The formula related to pH is defined as:

  • $\text{pH} = - \text{log}_{10}([H^+])$ where [H+] is the concentration of hydrogen ions in the solution.

• Practice Problem: Calculating pH

  • Problem: Find the pH of a solution where $[H^+] = 1 \times 10^{-4}$.

  • Step 1: Plug the value into the formula: $\text{pH} = -\text{log}_{10}(10^{-4})$.

  • Step 2: Apply the power rule: $\text{pH} = -(-4) \times \text{log}_{10}(10)$.

  • Solution: Since $\text{log}_{10}(10) = 1$, $\text{pH} = 4$. The substance is acidic.

Logarithmic Properties

Product Rule for Logarithms

• Logarithm and exponential functions are inverses, and thus they share similar properties.

• Product Rule: The logarithm of a product is equal to the sum of the logarithms of its factors:

  • $\text{log}<em>a(b \times c) = \text{log}</em>a(b) + \text{log}_a(c)$

Quotient Rule for Logarithms

• The quotient rule for logarithms states:

  • Logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator:

  • $\text{log}<em>a \bigg( \frac{b}{c} \bigg) = \text{log}</em>a(b) - \text{log}_a(c)$

Power Rule for Logarithms

• The power rule states that the log of a power is equal to the exponent times the log of the base:

  • $\text{log}<em>a(b^n) = n \times \text{log}</em>a(b)$

Expanding Logarithmic Expressions

• To expand logarithmic expressions:

  1. Apply the product rule for logs to express sums.

  2. Apply the quotient rule for logs to express differences.

  3. Apply the power rule to deal with exponents.

• Practice Problem: Expanding Logs

  • Problem: Expand $\text{log}_3(x^5y^2)$.

  • Step 1: Use the product rule: $\text{log}<em>3(x^5) + \text{log}</em>3(y^2)$.

  • Step 2: Use the power rule to move exponents to the front.

  • Solution: $5\text{log}<em>3(x) + 2\text{log}</em>3(y)$.

Condensing Logarithmic Expressions

• To condense sums, differences, or products of logarithms into a single logarithm:

  1. Apply the power property first, converting products of factors into logarithms of powers.

  2. Next, apply the product property for summation of logarithms.

  3. Finally, apply the quotient property for differences.

• Practice Problem: Condensing Logs

  • Problem: Condense $3\ln(x) - \ln(y)$.

  • Step 1: Apply the power rule: $\ln(x^3) - \ln(y)$.

  • Step 2: Apply the quotient rule.

  • Solution: $\ln\bigg(\frac{x^3}{y}\bigg)$.

Change-of-Base Formula for Logarithms

• Change-of-base Formula: Used for evaluating logarithms with bases other than common (10) or natural (e).

  • For any positive a and b (where a, b > 0):

  • $\text{log}<em>a(b) = \frac{\text{log}</em>{10}(b)}{\text{log}<em>{10}(a)}$ or $\text{log}</em>a(b) = \frac{\text{ln}(b)}{\text{ln}(a)}$

• Practice Problem: Change-of-Base

  • Problem: Evaluate $\text{log}_2(10)$ using the natural logarithm.

  • Step 1: Setup the ratio: $\frac{\ln(10)}{\ln(2)}$.

  • Step 2: Calculate: $\approx \frac{2.3025}{0.6931}$.

  • Solution: $\approx 3.3219$.