Logarithms

Logarithms

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Core definition: If $b^y = x$, then $y = \log_b(x)$ — logarithms are the inverse operation of exponentiation. The argument $x$ must always be positive ($x > 0$) and base $b > 0, b \neq 1$.

Must-know formulas:

• $\log_b(mn) = \log_b(m) + \log_b(n)$
• $\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)$
• $\log_b(m^k) = k \cdot \log_b(m)$
• Change of base: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$

3 high-yield CAT pointers:

• Domain trap: $\log_b(x)$ is undefined for $x \leq 0$ — always check argument positivity.
• Inequality flips: When comparing $\log_a(b)$ vs $\log_a(c)$ with $0 < a < 1$, the inequality sign reverses.
• Equation solving: $\log_b(f(x)) = k$ means $b^k = f(x)$, not $f(x) = k$.

---

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Definition and Base Restrictions

A logarithm answers the question: “To what power must base $b$ be raised to get $x$?” Formally, $y = \log_b(x)$ if and only if $b^y = x$. The domain of any logarithmic function is $(0, \infty)$ — no argument can be zero or negative. The range is $(-\infty, +\infty)$. Two base conditions are absolute: $b > 0$ and $b \neq 1$. If $b > 1$, the function is increasing; if $0 < b < 1$, it is decreasing.

Fundamental Properties

The three product/quotient/power rules form the backbone of log manipulation:

Rule	Statement
Product	$\log_b(mn) = \log_b m + \log_b n$
Quotient	$\log_b(m/n) = \log_b m - \log_b n$
Power	$\log_b(m^k) = k \cdot \log_b m$

Derived identities include $\log_b(1) = 0$, $\log_b(b) = 1$, and the crucial change-of-base formula: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$, which lets you evaluate logs in any base using a calculator (typically base 10 or base $e$).

Solving Logarithmic Equations

When $\log_b(f(x)) = \log_b(g(x))$ with same base $b$, you can equate arguments: $f(x) = g(x)$. However, both arguments must remain positive — solve for $x$, then discard any roots making $f(x) \leq 0$ or $g(x) \leq 0$. For $\log_b(f(x)) = k$ (where $k$ is a constant), convert to exponential form: $f(x) = b^k$.

Common CAT Patterns

• Simplifying complex expressions using log properties before attempting other operations.
• Finding number of digits in a large number $N$: characteristic of $\log_{10}(N) + 1$.
• Logarithmic inequalities require careful base-direction tracking.

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Formal Mechanisms

The identity $b^{\log_b(a)} = a$ holds universally for valid $a > 0, b > 0, b \neq 1$, but the converse $\log_b(b^x) = x$ reveals a subtle constraint: $x$ can be any real number because the exponential function accepts all reals as exponents. This explains why $\log_2(3)$ is perfectly defined (approximately 1.585) despite having no simple fraction representation — the result is irrational.

A critical distinction: logarithmic growth is slower than any positive power of $x$. As $x \to \infty$, $\frac{\log x}{x^p} \to 0$ for any $p > 0$. This fact appears in limits and comparing function growth rates, common in CAT’s more advanced algebra questions.

The Reciprocal Relationship Trap

The identity $\log_a(b) = \frac{1}{\log_b(a)}$ is frequently misapplied. Students often incorrectly assume $a^{\log_b(c)} = c^{\log_b(a)}$ without verifying — but this identity actually holds: $a^{\log_b(c)} = c^{\log_b(a)}$. The proof uses change-of-base: $a^{\log_b(c)} = a^{\frac{\ln c}{\ln b}} = (e^{\ln a})^{\frac{\ln c}{\ln b}} = e^{\ln a \cdot \frac{\ln c}{\ln b}} = e^{\ln c \cdot \frac{\ln a}{\ln b}} = c^{\log_b(a)}$.

Number of Digits via Logarithm

For a positive integer $N$, the number of digits $d$ satisfies $d - 1 \leq \log_{10}(N) < d$, so $d = \lfloor \log_{10}(N) \rfloor + 1$. Example: $\log_{10}(999) \approx 2.9996$, so $d = 2 + 1 = 3$ digits.

Common Mistakes to Avoid

1. Never write $\log(m + n) = \log m + \log n$ — this is false. Only products split.
2. Never write $\log(m/n) = \frac{\log m}{\log n}$ — the correct form is $\log m - \log n$.
3. When comparing $\log_a(b)$ across different bases, inequality direction flips if $0 < a < 1$.
4. Dropping the base in $\log_b(f(x)) = k$: the right side is $k$, not $f(x)$.

Practice Prompts

1. Solve for $x$: $\log_2(x + 1) + \log_2(x - 3) = 3$.
2. If $\log_3(2) = a$, express $\log_6(8)$ in terms of $a$.

Content adapted based on your selected roadmap duration. Switch tiers using the selector above.