Unit 2: Logarithms and Logarithmic Functions

A logarithm is the inverse operation of exponentiation. If \(a^y = x\), then \(\log_a x = y\), where:

• \(a > 0\), \(a \neq 1\)
• \(x > 0\)

1. Relationship with Exponents:
   • Exponential form: \(a^y = x\)
   • Logarithmic form: \(\log_a x = y\)
2. Common Logarithm:
   • Base \(10\), written as \(\log x\)
3. Natural Logarithm:
   • Base \(e\) (\(e \approx 2.718\)), written as \(\ln x\)

1. Product Rule:

   \[ \log_a (MN) = \log_a M + \log_a N \]

   Example: \(\log_3 (2 \times 4) = \log_3 2 + \log_3 4\).

2. Quotient Rule:

   \[ \log_a \left(\frac{M}{N}\right) = \log_a M - \log_a N \]

   Example: \(\log_2 (8 / 4) = \log_2 8 - \log_2 4\).

3. Power Rule:

   \[ \log_a (M^p) = p \cdot \log_a M \]

   Example: \(\log_5 (3^2) = 2 \cdot \log_5 3\).

4. Change of Base Formula:

   \[ \log_a M = \frac{\log_b M}{\log_b a} \]

   Example: \(\log_4 8 = \frac{\log 8}{\log 4}\).

   Substitute: \(\log 8 \approx 0.90309\) and \(\log 4 \approx 0.60206\):

   \[ \log_4 8 = \frac{0.90309}{0.60206} \approx 1.5. \]

Example 3: Using the Change of Base Formula

Evaluate \(\log_5 100\) using base \(10\):

1. Apply the Change of Base Formula:

\[ \log_5 100 = \frac{\log 100}{\log 5} \]

2. Substitute the known values (\(\log 100 = 2\) and \(\log 5 \approx 0.69897\)):

\[ \log_5 100 = \frac{2}{0.69897} \approx 2.86 \]

• The graph of \(y = \log_a x\) is the inverse of \(y = a^x\).
• Key features:
  • \(x > 0\)
  • Passes through \((1, 0)\)
  • Vertical asymptote at \(x = 0\)
  • Increases if \(a > 1\), decreases if \(0 < a < 1\)

1. Earthquake Magnitude (Richter Scale):

   Magnitude = \(\log_{10} \left(\frac{\text{Amplitude}}{\text{Reference Amplitude}}\right)\).

2. Population Growth:

   Solve for time in the exponential growth model:

   \[ A = P(1 + r)^t \implies t = \frac{\log(A/P)}{\log(1 + r)} \]

• The base \(a\) of a logarithm must be positive and not equal to 1.
• Logarithms are defined only for positive real numbers (\(x > 0\)).
• The notation \(\log_a x\) does not mean \(\frac{1}{\log x}\).
• Understanding logarithmic laws is crucial for simplifying and solving logarithmic and exponential equations.
• Logarithms can be used to solve for variables in exponential growth and decay problems.

1. Convert \(\log_4 64\) to exponential form.
2. Solve for \(x\): \(\log_2 x = 5\).
3. Evaluate \(\log_{10} 1000\).
4. Using the Change of Base Formula, calculate \(\log_7 49\).
5. Simplify \(\log_3 (81)\) using the Power Rule.
6. Solve for \(x\): \(\log_5 (x) = 3\).
7. Simplify \(\log_7 7 + \log_7 49\).
8. Simplify \(\log_2 \left(\frac{32}{4}\right)\) using the Quotient Rule.
9. Express \(\log_4 (16)\) using the Power Rule.
10. Calculate \(\log_{10} 1\).

1. Convert \(\log_6 36\) to exponential form.
2. Solve for \(x\): \(\log_4 x = 3\).
3. Evaluate \(\ln e^3\).
4. Using the Change of Base Formula, calculate \(\log_3 81\).
5. Simplify \(\log_5 (25)\) using the Power Rule.
6. Solve for \(x\): \(\log_{10} (x) = 2\).
7. Simplify \(\log_7 7 + \log_7 49\).
8. Simplify \(\log_2 \left(\frac{32}{4}\right)\) using the Quotient Rule.
9. Express \(\log_4 (16)\) using the Power Rule.
10. Calculate \(\log_{10} 1\).

• A logarithm is the inverse of an exponential function, defined as \(\log_a x = y\) if and only if \(a^y = x\).
• Logarithmic laws (Product, Quotient, Power, and Change of Base) are essential for simplifying and solving logarithmic equations.
• The common logarithm has a base of 10 (\(\log x\)), and the natural logarithm has a base of \(e\) (\(\ln x\)).
• Graphically, \(y = \log_a x\) is the inverse of \(y = a^x\), featuring a vertical asymptote at \(x = 0\) and passing through the point \((1, 0)\).
• Logarithms have practical applications in various fields, including measuring earthquake magnitudes and modeling population growth.