Concept of logarithms and their properties

1. Defining logarithms as the inverse operation of exponentiation and interpreting them as a 'scale of magnitude'

Logarithms as Inverse Operations

A logarithm $\log_b(x)$ answers the question: "To what power must base $b$ be raised to produce $x$?" Formally, $y = \log_b(x)$ if and only if $b^y = x$, where $b > 0$, $b \neq 1$, and $x > 0$. This defines logarithms as the inverse operation of exponentiation.

Logarithms compress large ranges of values into manageable scales, making them ideal for measuring orders of magnitude. A change of 1 in the logarithmic value corresponds to multiplication by the base in the original quantity.

Core Rules:

• $\log_b(b^y) = y$ for all real $y$
• $b^{\log_b(x)} = x$ for all $x > 0$
• $\log_b(1) = 0$ because $b^0 = 1$
• $\log_b(b) = 1$ because $b^1 = b$

This inverse relationship means logarithmic and exponential functions reflect across the line $y = x$.

Example: $\log_2(8) = 3$ because $2^3 = 8$.

2. Strict domain awareness and graphing the logarithmic function

Domain and Graph of Logarithmic Functions

The function $f(x) = \log_b(x)$ is defined only for $x > 0$. The domain is $(0, \infty)$ and the range is $(-\infty, \infty)$. The graph has a vertical asymptote at $x = 0$, meaning $\log_b(x) \to -\infty$ as $x \to 0^+$.

The graph passes through $(1, 0)$ and increases without bound as $x$ increases. For $b > 1$, the function is strictly increasing; for $0 < b < 1$, it is strictly decreasing.

Core Rules:

• Domain: $x > 0$ (strictly positive)
• Vertical asymptote: $x = 0$
• $x$-intercept: $(1, 0)$
• No $y$-intercept (undefined at $x = 0$)

Attempting to evaluate $\log_b(x)$ for $x \leq 0$ produces no real output.

Example: $\log_{10}(0.01) = -2$ because $10^{-2} = 0.01$.

3. Applying transformations to logarithmic graphs and observing domain shifts

Transformations of Logarithmic Graphs

Transformations of $f(x) = \log_b(x)$ follow standard function transformation rules. The function $g(x) = \log_b(x - h) + k$ shifts the graph horizontally by $h$ and vertically by $k$. Critically, horizontal shifts change the domain.

For $g(x) = \log_b(x - h)$, the domain becomes $x > h$ (not $x > 0$), and the vertical asymptote moves to $x = h$. Vertical shifts do not affect the domain or asymptote position.

Core Rules:

• Horizontal shift by $h$: domain becomes $x > h$, asymptote at $x = h$
• Vertical shift by $k$: range unchanged, $x$-intercept moves
• Reflection over $x$-axis: $-\log_b(x)$
• Vertical stretch/compression: $a\log_b(x)$ where $a \neq 0$

Always determine the new domain before graphing transformed logarithmic functions.

Example: $f(x) = \log_2(x - 3)$ has domain $x > 3$ and asymptote $x = 3$.

4. Core logarithmic properties and the Change of Base formula

Logarithmic Properties and Change of Base

Logarithms satisfy three fundamental algebraic properties derived from exponent laws. These properties enable simplification of complex logarithmic expressions.

Core Rules:

• Product Rule: $\log_b(MN) = \log_b(M) + \log_b(N)$ for $M, N > 0$
• Quotient Rule: $\log_b(M/N) = \log_b(M) - \log_b(N)$ for $M, N > 0$
• Power Rule: $\log_b(M^p) = p \cdot \log_b(M)$ for $M > 0$, any real $p$
• Change of Base: $\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$ for any valid base $k$

The Change of Base formula allows conversion between bases, essential for calculator evaluation (typically using base 10 or $e$).

These properties apply only when all arguments are positive.

Example: $\log_2(32) = \log_2(2^5) = 5 \cdot \log_2(2) = 5$.

5. Applications in pH scale, Richter scale, and decibels

Real-World Logarithmic Scales

Logarithmic scales compress exponential variations into linear measurements. The pH scale measures acidity: $\text{pH} = -\log_{10}[H^+]$, where $[H^+]$ is hydrogen ion concentration in moles per liter. A pH decrease of 1 represents a tenfold increase in acidity.

The Richter scale quantifies earthquake magnitude: $M = \log_{10}(A/A_0)$, where $A$ is amplitude and $A_0$ is a reference. Each integer increase represents 10 times greater amplitude.

Decibels (dB) measure sound intensity: $L = 10\log_{10}(I/I_0)$, where $I$ is intensity and $I_0$ is the threshold of hearing.

Core Rules:

• Each unit change represents a multiplicative change in the original quantity
• These scales handle values spanning many orders of magnitude
• All require $x > 0$ in the logarithmic argument

Example: pH 3 is 10 times more acidic than pH 4.