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Indices and Logarithms

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 10 Pathway: N/A

First Principles

Objective: Apply laws of indices and logarithms to simplify expressions and solve exponential and logarithmic equations.

Power Ladder & Logarithm Mirror

Select a base and move the target value slider to explore how exponents and logarithms invert each other.

INDEX FORM
LOGARITHMIC FORM

Step 1 — Concrete Growth: The Repeated Multiplier

Imagine a small agribusiness cooperative in Eldoret that doubles its seed inventory every harvest cycle. In cycle 0, they have \(1\) bag. In cycle 1, \(2^1 = 2\) bags; cycle 2, \(2^2 = 4\) bags; cycle 3, \(2^3 = 8\) bags. An index or exponent is simply a counter of repeated multiplication.

The Index Concept: An exponent \(n\) in \(a^n\) tracks how many times base \(a\) is multiplied by itself: \[ a^n = \underbrace{a \times a \times \dots \times a}_{n \text{ times}} \]

Step 2 — Geometric Dimension and Scale

Consider a unit square with side \(a\) (area \(a^2\)) versus a unit cube with side \(a\) (volume \(a^3\)). When we multiply \(a^2 \times a^3\), we are combining \(2\) factors of \(a\) with \(3\) factors of \(a\), giving \(a^{2+3} = a^5\). Division cancels repeated factors: \(\frac{a^5}{a^2} = a^{5-2} = a^3\).

Step 3 — Logarithms as the Inverse Operation

If powers move us forward along growth (e.g., "What is \(2^5\)? Answer: \(32\)"), logarithms reverse the question: "To what power must base \(2\) be raised to reach \(32\)?"

We write this relationship as the Fundamental Equivalence: \[ a^c = b \iff \log_a b = c \quad (a > 0, a \neq 1, b > 0) \]

Key Formulas

\[ a^{m} \times a^{n} = a^{m+n} \] Product Law of Indices: Add exponents when multiplying powers of the same base.
\[ \frac{a^{m}}{a^{n}} = a^{m-n} \quad (a \neq 0) \] Quotient Law of Indices: Subtract exponents when dividing powers of the same base.
\[ (a^{m})^{n} = a^{m \cdot n} \] Power of a Power: Multiply exponents when raising a power to another exponent.
\[ a^{-n} = \frac{1}{a^n} \quad \text{and} \quad a^{\frac{m}{n}} = \sqrt[n]{a^m} \] Negative and Fractional Indices: Negative represents reciprocals; fractional denominators represent roots.
\[ \log_a(m \cdot n) = \log_a m + \log_a n \] Product Law of Logarithms: Multiplication inside converts to addition outside.
\[ \log_a\left(\frac{m}{n}\right) = \log_a m - \log_a n \] Quotient Law of Logarithms: Division inside converts to subtraction outside.
\[ \log_a(m^k) = k \cdot \log_a m \] Power Law of Logarithms: An exponent inside acts as a coefficient multiplier in front of the log.
\[ \log_a b = \frac{\log_c b}{\log_c a} \] Change of Base Formula: Evaluates logs with any base using a common base (e.g., base 10 or \(e\)).

Worked Examples

Example 1 (Easy): Simplifying Index Expressions

Problem: Simplify \(\frac{3^6 \times 3^4}{3^7}\) and express the answer as a single integer.

  1. Combine the numerator using the Product Law:
    \[ 3^6 \times 3^4 = 3^{6 + 4} = 3^{10} \]
  2. Apply the Quotient Law with the denominator:
    \[ \frac{3^{10}}{3^7} = 3^{10 - 7} = 3^3 \]
  3. Evaluate the numerical result:
    \[ 3^3 = 3 \times 3 \times 3 = 27 \]

Final Answer: \(27\)

Example 2 (Medium): Solving an Exponential Equation

Problem: Solve for \(x\) in the equation \(4^{2x - 1} = 64\).

  1. Express both sides in terms of a common base (Base 4 or Base 2):
    Notice that \(64 = 4^3\) (or \(2^6\)). Let us use base 4:
    \[ 4^{2x - 1} = 4^3 \]
  2. Equate the exponents since the bases are identical:
    \[ 2x - 1 = 3 \]
  3. Solve the linear equation for \(x\):
    \[ 2x = 3 + 1 = 4 \implies x = 2 \]

Final Answer: \(x = 2\)

Example 3 (Hard): Multi-Step Logarithmic Equation

Problem: Solve for \(x\): \(\log_{10}(x + 3) + \log_{10}(x - 3) = 1 + \log_{10}(1.6)\).

  1. Apply the product rule to the left-hand side:
    \[ \log_{10}[(x + 3)(x - 3)] = \log_{10}(x^2 - 9) \]
  2. Express 1 on the right-hand side in base 10:
    Since \(1 = \log_{10}(10)\), combine the right side using the product rule:
    \[ \log_{10}(10) + \log_{10}(1.6) = \log_{10}(10 \times 1.6) = \log_{10}(16) \]
  3. Set arguments equal:
    \[ \log_{10}(x^2 - 9) = \log_{10}(16) \implies x^2 - 9 = 16 \]
  4. Solve for \(x\):
    \[ x^2 = 25 \implies x = \pm 5 \]
  5. Verify domain restrictions: For \(\log_{10}(x - 3)\), \(x - 3 > 0 \implies x > 3\). Thus, \(x = -5\) is extraneous. The only valid solution is \(x = 5\).

Final Answer: \(x = 5\)

Common Mistakes

Misconception 1: Distributing Logs over Addition

Incorrect \(\log_a(x + y) = \log_a x + \log_a y\)

Correction The logarithm of a sum cannot be broken down! The product rule states that \(\log_a(x \cdot y) = \log_a x + \log_a y\).

Why it feels right We are used to linear distributive properties such as \(c(x + y) = cx + cy\). However, logarithms represent exponents, which convert multiplication into addition, not addition into addition.

Misconception 2: Subtracting Exponents in Bases Multiplied

Incorrect \(2^5 \times 2^3 = 2^{15}\) or \(2^5 \times 2^3 = 4^8\)

Correction \(2^5 \times 2^3 = 2^{5+3} = 2^8\). The base remains unchanged while exponents are added.

Why it feels right Learners sometimes multiply the base (getting 4) or multiply the powers because of the multiplication sign (\(\times\)).

Misconception 3: Forgetting Domain Constraints in Log Equations

Incorrect Accepting negative root solutions blindly without checking \(x > 0\) in original log terms.

Correction The argument of a logarithm must always be strictly positive (\(u > 0\) for \(\log_a u\)). Any algebraic solution that yields a non-positive argument must be discarded as extraneous.

Real World

1. SACCO Savings & The Doubling Law (Compound Interest)

A tea farmers' cooperative in Kericho offers a savings scheme compounded annually at 7.2%. The time \(t\) in years required to double an investment is derived via logarithms: \[ P(1 + 0.072)^t = 2P \implies (1.072)^t = 2 \implies t = \frac{\log 2}{\log 1.072} \approx 10 \text{ years} \]

2. Mobile Network Traffic and Data Packets

Telecommunications engineers in Nairobi measure signal attenuation in decibels (dB). Decibels use base-10 logarithms: \[ \text{dB} = 10 \log_{10}\left(\frac{P_{\text{out}}}{P_{\text{in}}}\right) \] A signal loss factor of 100 equates to a drop of \(10 \log_{10}(100) = 20\text{ dB}\).

3. Rift Valley Seismic Activity (Richter Scale)

Geologists studying tremors along the East African Rift use the logarithmic Richter scale: \[ M = \log_{10}\left(\frac{I}{S}\right) \] An earthquake of magnitude 6 has an amplitude \(10^{6-4} = 10^2 = 100\) times greater than an earthquake of magnitude 4.

Practice

Simplify the expression \(\frac{2^5 \times 2^3}{2^4}\). What is the exact integer value? (Type only the number, e.g., 42)
Review the concepts above.
Evaluate the exact value of \(\log_{3}(81)\). (Type only the number, e.g., 42)
Review the concepts above.
An agricultural research team in Nakuru determines that a crop microbe population triples every hour according to the equation \(3^{2x - 1} = 243\). Solve for \(x\). (Type only the number, e.g., 42)
Review the concepts above.
A financial model calculates an index factor \(K\) as \(K = \log_{10}(25) + \log_{10}(400) - \log_{10}(10)\). Calculate the value of \(K\). (Type only the number, e.g., 42)
Review the concepts above.
Solve for \(y\) in the exponential equation \(9^{y+1} = 27^{y-1}\). (Type only the number, e.g., 42)
Review the concepts above.
Find the positive solution for \(x\) in the logarithmic equation: \(\log_{2}(x + 6) + \log_{2}(x - 6) = 6\). (Type only the number, e.g., 42)
Review the concepts above.