5.2 Logarithmic Functions and Properties

5.2 Logarithmic Functions and Properties

Because exponential functions $f(x) = b^x$ (where $b > 0, b \neq 1$) are strictly monotonic (either strictly increasing or strictly decreasing across their entire domain), they are one-to-one and possess inverses. The inverse of an exponential function is a logarithmic function. Logarithms allow us to isolate and solve for unknown exponents, making them indispensible across mathematics and scientific fields.

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1. Definition of Logarithmic Functions as Inverses

For any base $b > 0$ and $b \neq 1$, the logarithmic function with base $b$, denoted by $f(x) = \log_b(x)$, is defined as:

$y = \log_b(x) \iff b^y = x, \quad \text{for } x > 0$

In prose: "$\log_b(x)$ is the exponent $y$ to which the base $b$ must be raised to produce the value $x$."

Fundamental Inverse Identities

Because $f(x) = b^x$ and $g(x) = \log_b(x)$ are inverse functions, their compositions yield the identity function:

1. $\log_b(b^x) = x$ for all $x \in (-\infty, \infty)$
2. $b^{\log_b(x)} = x$ for all $x > 0$

Graphical Relationship and Characteristics

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2. Common and Natural Logarithms

Two logarithmic bases are standardized in mathematics and scientific notation:

1. Common Logarithm (Base $10$): Written without an explicit base subscript: $\log(x) = \log_{10}(x)$ Used in measuring physical scales like sound intensity (decibels), earthquake magnitude (Richter scale), and chemical acidity ($\text{pH} = -\log[\text{H}^+]$).

2. Natural Logarithm (Base $e$): Written using the Latin abbreviation $\ln$ (logarithmus naturalis): $\ln(x) = \log_e(x)$ The natural log obeys all standard logarithmic rules, with natural inverse properties: $\ln(e^x) = x \quad \text{and} \quad e^{\ln(x)} = x$

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3. Core Laws / Properties of Logarithms

Let $b > 0, b \neq 1$, and let $u, v > 0$ be positive real numbers.

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4. Expanding and Condensing Expressions

Mastery of algebraic manipulation using log properties is heavily tested on the CLEP Precalculus exam.

Worked Example 1: Expanding a Complex Logarithmic Expression

Expand the logarithmic expression completely into sums, differences, and constant multiples of logarithms:

$\ln \left( \frac{x^3 \sqrt{y + 2}}{z^5} \right), \quad \text{where } x > 0, \, y > -2, \, z > 0$

Solution:

1. Apply the Quotient Property to separate the numerator and denominator: $\ln \left( \frac{x^3 \sqrt{y + 2}}{z^5} \right) = \ln(x^3 \sqrt{y + 2}) - \ln(z^5)$
2. Apply the Product Property to the numerator product: $= \ln(x^3) + \ln(\sqrt{y + 2}) - \ln(z^5)$
3. Rewrite radical powers as rational exponents ($\sqrt{y + 2} = (y + 2)^{1/2}$): $= \ln(x^3) + \ln\left((y + 2)^{1/2}\right) - \ln(z^5)$
4. Apply the Power Property to bring all exponents to the front: $= 3\ln(x) + \frac{1}{2}\ln(y + 2) - 5\ln(z)$

Worked Example 2: Condensing into a Single Logarithm

Condense the following expression into a single logarithm with a coefficient of 1:

$2 \log_3(x) + \frac{1}{3} \log_3(y) - 4 \log_3(z)$

Solution:

1. Apply the Power Property in reverse to move coefficients inside as exponents: $\log_3(x^2) + \log_3(y^{1/3}) - \log_3(z^4)$
2. Combine the positive additions using the Product Property: $\log_3(x^2 \cdot y^{1/3}) - \log_3(z^4)$
3. Apply the Quotient Property for the subtraction: $\log_3 \left( \frac{x^2 \sqrt[3]{y}}{z^4} \right)$

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5. Domain Restrictions and Vertical Asymptotes

Because the argument of a logarithm must be strictly positive ($> 0$), finding the domain of $f(x) = \log_b(g(x))$ requires solving the inequality:

$g(x) > 0$

The vertical asymptote occurs where the inner argument $g(x) = 0$.

Worked Example 3: Finding Domain and Vertical Asymptote

Determine the domain and vertical asymptote of $f(x) = \log_5(12 - 3x)$.

Solution:

1. Set the argument strictly greater than zero: $12 - 3x > 0 \implies 12 > 3x \implies x < 4$
2. In interval notation, the domain is $(-\infty, 4)$.
3. Set the argument equal to zero to find the vertical asymptote: $12 - 3x = 0 \implies x = 4$ The vertical asymptote is the vertical line $x = 4$.

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6. Change-of-Base Formula

The Change-of-Base Formula converts logarithms from any base $b$ to an arbitrary new positive base $a$ (most commonly base $10$ or base $e$):

$\log_b(x) = \frac{\log_a(x)}{\log_a(b)} = \frac{\ln(x)}{\ln(b)} = \frac{\log(x)}{\log(b)}$

Worked Example 4: Exact Logarithmic Evaluation

Evaluate $\log_{16}(32)$ using the Change-of-Base Formula.

Solution: Choose base $2$ (since both $16$ and $32$ are powers of $2$): $\log_{16}(32) = \frac{\log_2(32)}{\log_2(16)} = \frac{5}{4} = 1.25$

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7. CLEP Exam Traps & Common Errors

• Trap 1: Inventing Non-Existent Log Rules
  • $\log_b(u + v) \neq \log_b(u) + \log_b(v)$. (Log of a sum CANNOT be split!)
  • $\frac{\log_b(u)}{\log_b(v)} \neq \log_b(u - v)$. (Quotient of two logs is NOT the log of a difference!)
• Trap 2: Confusing $\log_b(u^p)$ with $(\log_b(u))^p$
  The power rule $p \log_b(u)$ ONLY applies when the exponent $p$ belongs to the argument $u$, not when the entire logarithmic expression is raised to a power.
• Trap 3: Logarithms of Non-Positive Numbers
  Evaluating $\log_b(0)$ or $\log_b(\text{negative})$ within real numbers is undefined. Always check domain boundaries.