Indices and Logarithms

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Indices (Exponents) are a shorthand way of writing repeated multiplication. Logarithms are the inverse operation of indices.

Index Notation: $$a^n = \underbrace{a \times a \times a \times … \times a}_{n \text{ times}}$$

Where $a$ is the base and $n$ is the index/exponent.

Laws of Indices:

Law	Formula	Example
Product	$a^m \times a^n = a^{m+n}$	$2^3 \times 2^4 = 2^7$
Quotient	$a^m \div a^n = a^{m-n}$	$2^5 \div 2^3 = 2^2$
Power of power	$(a^m)^n = a^{mn}$	$(2^3)^2 = 2^6$
Zero index	$a^0 = 1$ (for $a \neq 0$)	$5^0 = 1$
Negative index	$a^{-n} = \frac{1}{a^n}$	$2^{-3} = \frac{1}{8}$
Fractional index	$a^{\frac{1}{n}} = \sqrt[n]{a}$	$8^{\frac{1}{3}} = 2$

Logarithm Definition: $$\log_a x = y \quad \Leftrightarrow \quad a^y = x$$

“Log base $a$ of $x$ equals $y$” means “$a$ to the power $y$ equals $x$.”

Examples: $$\log_2 8 = 3 \quad \text{because } 2^3 = 8$$ $$\log_{10} 1000 = 3 \quad \text{because } 10^3 = 1000$$ $$\log_5 25 = 2 \quad \text{because } 5^2 = 25$$

⚡ WAEC Tip: Logarithms and indices are inverses. If $a^x = y$, then $\log_a y = x$. You can convert between forms!

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Laws of Logarithms:

Law	Formula
Product	$\log_a (xy) = \log_a x + \log_a y$
Quotient	$\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$
Power	$\log_a (x^n) = n \log_a x$
Change of base	$\log_a x = \frac{\log_b x}{\log_b a}$

Example — Using log laws: $$\log_2 8 + \log_2 4 = \log_2 (8 \times 4) = \log_2 32 = 5$$

Simplifying Expressions:

Problem: Simplify $\frac{2^5 \times 2^3}{2^4}$ $$= 2^{5+3-4} = 2^4 = 16$$

Problem: Simplify $\frac{27^{\frac{2}{3}}}{9^{\frac{1}{2}}}$ $$= \frac{(3^3)^{\frac{2}{3}}}{(3^2)^{\frac{1}{2}}} = \frac{3^2}{3^1} = \frac{9}{3} = 3$$

Solving Logarithmic Equations:

Problem: Solve $\log_2 x = 5$ $$x = 2^5 = 32$$

Problem: Solve $\log_3 (x + 1) = 4$ $$x + 1 = 3^4 = 81$$ $$x = 80$$

Problem: Solve $\log_2 x + \log_2 (x - 3) = 3$ $$\log_2 [x(x-3)] = 3$$ $$x(x-3) = 2^3 = 8$$ $$x^2 - 3x - 8 = 0$$ $$(x-4)(x+2) = 0$$ $$x = 4 \text{ or } x = -2$$

Since $x > 3$ (domain: $x - 3 > 0$ for $\log_2(x-3)$), $x = 4$.

Exponential Equations:

Problem: Solve $3^{x+1} = 27$ $$3^{x+1} = 3^3$$ $$x + 1 = 3$$ $$x = 2$$

Problem: Solve $2^x = 10$ Take $\log_{10}$ of both sides: $$\log 2^x = \log 10$$ $$x \log 2 = 1$$ $$x = \frac{1}{\log 2} \approx \frac{1}{0.3010} \approx 3.32$$

⚡ Common Student Mistakes: Confusing $\log(xy)$ with $\log x \times \log y$ — product rule is $\log(xy) = \log x + \log y$. Forgetting to check the domain of logarithmic functions (argument must be > 0). When changing bases, using the wrong base in the numerator.

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for thorough preparation.

Natural Logarithms: When the base is $e$ (Euler’s number, $e \approx 2.71828…$): $$\ln x = \log_e x$$

This is used extensively in calculus and exponential growth/decay problems.

Change of Base Formula: $$\log_a x = \frac{\ln x}{\ln a} = \frac{\log_b x}{\log_b a}$$

This allows you to calculate logs with any base using a calculator (which usually has $\log_{10}$ and $\ln$):

Problem: Evaluate $\log_2 7$ $$\log_2 7 = \frac{\ln 7}{\ln 2} = \frac{1.9459…}{0.6931…} \approx 2.807$$

Simultaneous Logarithmic Equations:

Problem: $$\log_2 x + \log_2 y = 5$$ $$x - y = 4$$

From (1): $\log_2(xy) = 5$, so $xy = 32$ From (2): $x = y + 4$

Substitute: $$(y+4)y = 32$$ $$y^2 + 4y - 32 = 0$$ $$(y+8)(y-4) = 0$$ $$y = 4 \quad (\text{since } y > 0 \text{ for } \log_2 y \text{ to exist})$$

Then $x = 8$

Solving with Indices:

Problem: Solve $4^{x+1} = 8^{x-1}$ $$(2^2)^{x+1} = (2^3)^{x-1}$$ $$2^{2x+2} = 2^{3x-3}$$ $$2x + 2 = 3x - 3$$ $$x = 5$$

Problem: Solve $3^{2x} - 10 \times 3^x + 9 = 0$

Let $y = 3^x$: $$y^2 - 10y + 9 = 0$$ $$(y-1)(y-9) = 0$$ $$y = 1 \text{ or } y = 9$$ $$3^x = 1 \Rightarrow x = 0$$ $$3^x = 9 \Rightarrow x = 2$$

Compound Interest: $$A = P\left(1 + \frac{r}{100}\right)^n$$

Problem: ₦50,000 invested at 8% per annum for 5 years. Find the amount. $$A = 50000\left(1 + \frac{8}{100}\right)^5 = 50000(1.08)^5 = 50000 \times 1.4693 = \text{₦73,465}$$

Population Growth: Problem: A town’s population of 100,000 grows at 3% per year. Find the population after 10 years. $$P = 100000(1 + 0.03)^{10} = 100000 \times 1.3439 = \text{134,390}$$

Exponential Decay: Problem: A radioactive substance has half-life of 6 years. If initial amount is 80 g, find amount after 20 years.

Half-life formula: $N = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T}}$ Where $T$ = half-life, $t$ = time elapsed

$$N = 80 \times \left(\frac{1}{2}\right)^{\frac{20}{6}} = 80 \times \left(\frac{1}{2}\right)^{\frac{10}{3}} = 80 \times 0.1976 = \text{15.8 g (approx)}$$

pH and Decibel Scales:

pH Scale (logarithmic, base 10): $$\text{pH} = -\log_{10}[H^+]$$

If $[H^+] = 2 \times 10^{-5}$: $$\text{pH} = -\log(2 \times 10^{-5}) = -( \log 2 + \log 10^{-5}) = -(0.301 - 5) = 4.699 \approx 4.7$$

Decibel Scale (logarithmic, base 10): $$\beta = 10 \log_{10}\left(\frac{I}{I_0}\right) \quad \text{(decibels)}$$

If intensity $I = 100 \times I_0$: $$\beta = 10 \log_{10}(100) = 10 \times 2 = 20 \text{ dB}$$

Richter Scale (earthquake magnitude): $$M = \log_{10}\left(\frac{A}{A_0}\right)$$

If amplitude $A = 1000 \times A_0$: $$M = \log_{10}(1000) = 3$$

Solving Inequalities with Indices and Logs:

Problem: Solve $2^{x+1} > 16$ $$2^{x+1} > 2^4$$ $$x + 1 > 4$$ $$x > 3$$

Problem: Solve $\log_2 x < 3$ $$x < 2^3 = 8$$ Also need $x > 0$ (domain) So: $0 < x < 8$

Evaluating Logarithms without a Calculator:

Problem: Find $\log_2 48$ $$\log_2 48 = \log_2 (16 \times 3) = \log_2 16 + \log_2 3 = 4 + \log_2 3$$

Since $\log_2 3$ doesn’t simplify nicely, the answer is $4 + \log_2 3$.

Or using change of base with known values: $$\log_2 3 = \frac{\log_{10} 3}{\log_{10} 2} = \frac{0.4771}{0.3010} \approx 1.585$$ $$\log_2 48 \approx 4 + 1.585 = 5.585$$

⚡ WAEC Examination Patterns: Apply laws of indices to simplify expressions. Solve exponential and logarithmic equations. Solve problems involving compound interest and exponential growth/decay. Use change of base formula. Solve simultaneous equations with logs. Apply logs to real-world scales (pH, decibels, Richter).