Indices and Logarithms

Indices and Logarithms

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Indices (exponents) tell you how many times a base a multiplies itself: $a^n = a \times a \times \cdots \times a$ (n times). Logarithms are the inverse: if $a^x = N$, then $x = \log_a N$, valid only for $a>0$, $a \neq 1$, $N>0$.

Must-know rules:

• Multiplication: $a^m \times a^n = a^{m+n}$
• Division: $a^m \div a^n = a^{m-n}$
• Power of a power: $(a^m)^n = a^{mn}$
• Zero index: $a^0 = 1$ (for $a \neq 0$)
• Log product/quotient/power: $\log_a(MN)=\log_a M + \log_a N$; $\log_a(M/N)=\log_a M - \log_a N$; $\log_a(M^n)=n\log_a M$

NABTEB pointers: expect 1–3 objective items; simplify with negative/fractional indices; read four-figure log tables using characteristic + mantissa; convert between $\log_{10}$, $\ln$, and $\log_a$ via change of base.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Laws of Indices

For $a,b>0$ and real numbers $m,n$:

Rule	Formula
Multiplication	$a^m \times a^n = a^{m+n}$
Division	$a^m \div a^n = a^{m-n}$
Power of a power	$(a^m)^n = a^{mn}$
Power of a product	$(ab)^n = a^n b^n$
Zero index	$a^0 = 1$
Negative index	$a^{-n} = \dfrac{1}{a^n}$
Fractional index	$a^{m/n} = \sqrt[n]{a^m}$

These collapse multi-term expressions into single powers—essential for NABTEB simplification items.

Definition of a Logarithm

The logarithm is the inverse of exponentiation. If $a^x = N$ then $x = \log_a N$. Two special cases follow directly: $\log_a a = 1$ (because $a^1 = a$) and $\log_a 1 = 0$ (because $a^0 = 1$). The base restriction $a \neq 1$ matters because any number raised to $1$ is itself, so $\log_1 N$ would be undefined for $N \neq 1$.

Laws of Logarithms

These mirror the index laws because logs are inverses:

• Product: $\log_a(MN) = \log_a M + \log_a N$
• Quotient: $\log_a(M/N) = \log_a M - \log_a N$
• Power: $\log_a(M^n) = n \log_a M$

Worked: expand $\log_2(8 \times 5^2) = \log_2 8 + 2\log_2 5 = 3 + 2\log_2 5$.

Common Patterns in NABTEB

1. Simplify $2^3 \times 2^{-5} \times 4^2 = 2^{3-5+4} = 2^2 = 4$.
2. Express $27^{2/3}$ as $\sqrt[3]{27^2} = 9$.
3. Solve $3^{x+1} = 81$: since $81 = 3^4$, $x+1=4 \Rightarrow x=3$.
4. Read $\log_10 273.5$ from tables: characteristic $2$, mantissa from row 273/column 5, then find antilog to recover the number.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Change of Base and Log Tables

To evaluate $\log_a N$ with four-figure tables (which tabulate $\log_{10}$), use $$\log_a N = \frac{\log_{10} N}{\log_{10} a}.$$ On NABTEB essay papers you will be expected to read $\log_{10} x$ in two parts: the characteristic (the integer part, determined by where the decimal sits: e.g. $\log 2735 = 3 + \log 2.735$) and the mantissa (the fractional part, looked up in the table). The number is recovered via the antilogarithm table. Mixing these up is the single most common cause of lost marks.

Edge Cases and Common Mistakes

• $(a+b)^n \neq a^n + b^n$; only $(ab)^n = a^n b^n$.
• $\log(M+N) \neq \log M + \log N$; the law applies to products, not sums.
• Negative inputs are illegal: $\log_a N$ exists only for $N>0$.
• $a^0 = 1$, never $0$, even when $a$ itself is a fraction.
• Fractional indices $a^{1/2}=\sqrt{a}$ assume $a \geq 0$ in real arithmetic; $a^{3/2} = a\sqrt{a}$ requires $a \geq 0$.

Connections

Logarithms linearise exponential growth, which is why they appear in pH ($\log_{10}[H^+]$), decibel scales ($10\log_{10}$), and compound-interest decay: $A = Pe^{rt}$ becomes $\ln A = \ln P + rt$. Recognising this link helps in NABTEB’s integrated word problems.

Practice Prompts

1. Without tables, simplify $\dfrac{9^{1/2} \times 27^{-1/3}}{(\tfrac{1}{3})^{-2}}$.
2. Given $\log_5 2 = 0.4307$ and $\log_5 3 = 0.6826$, find $\log_5 60$.

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