Logarithms Explained

Logarithms are a fundamental mathematical concept that express the power to which a base number must be raised to yield a given value.

Key Concepts and Definitions

• Definition of Logarithms: The expression $a^y = x$ corresponds to $y = \log_a(x)$.
  • Examples:
    • $2^3 = 8$ implies $\log_2(8) = 3$.
    • $10^2 = 100$ implies $\log_{10}(100) = 2$.

• Importance of Base Conditions:
  • Base $a > 0$: Ensures expressions are valid in the real number domain, maintaining positivity.
  • Base $a \neq 1$: Prevents trivial outcomes, ensuring logarithmic relationships hold significance.

Logarithmic Laws

Understanding these laws is essential when manipulating logarithmic expressions:

• Product Law of Logarithms: $\log_a(xy) = \log_a(x) + \log_a(y)$.
• Quotient Law of Logarithms: $\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$.
• Power Law of Logarithms: $\log_a(x^k) = k \cdot \log_a(x)$.

Graphical Representations

• Logarithmic Curve Features:
  • Domain: $x > 0$.
  • Range: All real numbers.
  • Intercepts: The graph intersects the y-axis at $\log_a(1) = 0$.
  • Vertical Asymptotes: As $x$ approaches zero, a vertical asymptote is presented.

• Visual Activity: Draw a graph of $y = \log_a(x)$ and identify these key features.

Graph Comparisons and Inverse Relationships

• Inverse Relationships: The graph of a logarithm $y = \log_e(x)$ is reflected across the line $y = x$ from its exponential counterpart $y = e^x$.

Euler's Number 'e'

• Euler's Number $e$: Approximately 2.71828, it plays a crucial role in scenarios involving continuous compounding and natural growth.

Example Context: Continuous Compounding

• Formula for continuous compounding: $A = Pe^{rt}$

  • $A$: Accumulated amount.
  • $P$: Principal amount.
  • $r$: Annual interest rate (expressed as a decimal).
  • $t$: Time in years.

• Numerical Example:

  • Given $P = £1000$, $r = 5\%$ (or $0.05$), and $t = 3$ years:
    • Calculation:
    $$A = 1000e^{0.05 \times 3} \approx 1161.83$$
  • This demonstrates exponential growth via continuous compounding.

Natural Logarithms $\ln(x)$

• Definition: The natural logarithm with base-e simplifies processes in differentiation and integration.

Inverse Relationship

Simplification Examples

• Solve $\ln(e^3)$:
  • Result: $\ln(e^3) = 3$.
  • This illustrates the inverse function property effectively.

Graphs and Applications of Exponential and Logarithmic Functions

Key Features of Exponential Graphs

• Exponential Function Standard Form: $y = e^x$ or $y = ka^x + c$.
• Graph Characteristics:
  • Domain: All real numbers ($x \in \mathbb{R}$).
  • Range: $y > 0$.
  • Asymptotes: Horizontal at $y = 0$.
  • Intercept: $y$-intercept at $(0, 1)$ for $y = e^x$.

:::important Remember

• Domain: All real numbers ($x \in \mathbb{R}$)
• Range: $y > 0$
• Asymptote: Horizontal at $y = 0$
• $y$-intercept: $(0, 1)$ for $y = e^x$ :::

Graph Transformations

• Scaling:

  • Changing $k$ results in vertical stretching or compression.
  • Example: Compare $y = 2e^x$ to $y = e^x$ by plotting points for $x = 0, \pm1, \pm2$. Note the stretch in $y = 2e^x$ compared to $y = e^x$.

• Translation:

  • Altering $c$ shifts the graph vertically.
  • Example: Increasing $c$ results in an upward shift.

• Reflection:

  • Reflections across axes change the orientation of the graph.
  • Example: A negative coefficient results in reflection across the x-axis.

Real-world Applications

• Exponential functions are prevalent in contexts such as population growth and radioactive decay, demonstrating their practical relevance in mathematical concepts.