Logarithms

Core Concepts

1. Standard Form Standard form (or scientific notation) is a way of writing very large or very small numbers compactly. A number is expressed in standard form when it is written as: $$ A \times 10^n $$ where $1 \le A < 10$ and $n$ is an integer (positive, negative, or zero).

For numbers equal to or greater than 10, the decimal point is moved to the left, and $n$ is positive.
For numbers less than 1, the decimal point is moved to the right, and $n$ is negative.

2. Laws of Logarithms A logarithm is fundamentally an exponent. The mathematical statement $y = \log_a x$ answers the question "to what power $y$ must we raise the base $a$ to obtain $x$?" Logarithmic and index forms are interchangeable: $$ y = \log_a x \iff a^y = x $$ The fundamental laws of logarithms (where $x > 0$, $y > 0$, $a > 0$, and $a \neq 1$) are:

Product Law: $\log_a(xy) = \log_a x + \log_a y$
Quotient Law: $\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$
Power Law: $\log_a(x^n) = n \log_a x$
Identity and Zero Properties: $\log_a a = 1$ and $\log_a 1 = 0$
Change of Base: $\log_a b = \frac{\log_c b}{\log_c a}$

3. Tables of Logarithms Common logarithms are logarithms evaluated to base 10, frequently written without the base as $\log x$. The common logarithm of a number consists of two components:

Characteristic: The integer part, which can be positive, negative, or zero. It corresponds to the exponent $n$ when the number is written in standard form.
Mantissa: The fractional decimal part, which is strictly positive and is read from 4-figure logarithmic tables.

When solving multiplication, division, or fractional powers of numbers, operations are simplified into addition, subtraction, or multiplication of logarithms. The final result is obtained by reading the antilogarithm from the tables.

Worked Examples

Example 1: Standard Form Calculation Simplify the expression $\frac{7 \times 10^4}{0.000035}$ and hence write the answer in standard form.

Solution: First, express the decimal in the denominator in standard form: $$ 0.000035 = 3.5 \times 10^{-5} $$ Substitute this back into the original expression: $$ \frac{7 \times 10^4}{3.5 \times 10^{-5}} $$ Group the numeral coefficients and the powers of 10 separately: $$ = \left( \frac{7}{3.5} \right) \times \left( \frac{10^4}{10^{-5}} \right) $$ Divide the coefficients and apply the laws of indices to the powers of 10: $$ = 2 \times 10^{4 - (-5)} $$ $$ = 2 \times 10^{4 + 5} $$ $$ = 2 \times 10^9 $$

Example 2: Evaluating Logarithms using Given Values By using $\log 2 = 0.3010$ and $\log 3 = 0.4771$, find $\log 72$.

Solution: First, express 72 as a product of its prime factors: $$ 72 = 8 \times 9 = 2^3 \times 3^2 $$ Apply the logarithm to this prime factored form: $$ \log 72 = \log (2^3 \times 3^2) $$ Apply the Product Law of logarithms to split the terms: $$ \log 72 = \log (2^3) + \log (3^2) $$ Apply the Power Law of logarithms to bring the exponents down: $$ \log 72 = 3\log 2 + 2\log 3 $$ Substitute the given numerical values for $\log 2$ and $\log 3$: $$ \log 72 = 3(0.3010) + 2(0.4771) $$ $$ \log 72 = 0.9030 + 0.9542 $$ $$ \log 72 = 1.8572 $$

Example 3: Solving Logarithmic Equations Determine the value of $y$ in the equation $\log_{10}(3y+2) - 1 = \log_{10}(y-4)$.

Solution: Recall the identity property of logarithms base 10: $1 = \log_{10} 10$. Substitute this identity into the original equation: $$ \log_{10}(3y+2) - \log_{10} 10 = \log_{10}(y-4) $$ Apply the Quotient Law on the left-hand side to combine the logarithms: $$ \log_{10}\left( \frac{3y+2}{10} \right) = \log_{10}(y-4) $$ Since the bases are identical on both sides, their arguments must be equal: $$ \frac{3y+2}{10} = y - 4 $$ Multiply both sides by 10 to eliminate the fraction: $$ 3y + 2 = 10(y - 4) $$ $$ 3y + 2 = 10y - 40 $$ Group the terms containing $y$ on one side, and the constants on the other side: $$ 2 + 40 = 10y - 3y $$ $$ 42 = 7y $$ $$ y = 6 $$

NECTA Exam Focus

In the NECTA CSEE exams, Logarithms frequently appear in Section A (Basic skills). Analysis of recent past papers highlights a few consistent trends:

Decomposition using Prime Factors: A very common question type provides decimal approximations for $\log 2$, $\log 3$, and occasionally $\log 5$, requiring students to evaluate the logarithm of a larger composite number or a fraction (e.g., $\log 72$, $\log 40500$, or $\log 2\frac{1}{4}$). Success in these problems hinges on accurately finding the prime factorization and meticulously applying the product, quotient, and power laws.
Solving Logarithmic Equations: Students are frequently asked to solve for an unknown variable within an equation containing logarithms. A recurrent stumbling block is forgetting how to handle constants (like $1$, $2$, or $4$) that sit outside the log expressions. It is crucial to express the constant as a logarithm of the same base (e.g., rewriting $1$ as $\log_{10} 10$, or $4$ as $\log_3 3^4 = \log_3 81$) before applying the laws of logarithms to consolidate the terms.
Standard Form Arithmetic: Exams regularly test the ability to multiply or divide numbers in standard form. Be cautious when division results in a leading coefficient less than 1 or greater than 10 (e.g., $0.5 \times 10^n$ or $25 \times 10^n$); the decimal point and the exponent must be subsequently adjusted to satisfy the strict condition $1 \le A < 10$.

Practice Problems

Evaluate $\log_{10} 40500$ given that $\log_{10} 2 = 0.3010$, $\log_{10} 3 = 0.4771$ and $\log_{10} 5 = 0.6990$.
Given that $\log 2 = 0.3010$ and $\log 3 = 0.4771$, find the value of $\log\left(2\frac{1}{4}\right)$ without using mathematical tables.
Simplify the expression $\log_a \sqrt{a} + \log_a (a^2)$.
Solve for $x$ in the equation: $4 + 3\log_3 x = \log_3 24$.

Subtopics

Standard form
Laws of logarithms
Tables of logarithms

Crosswalk Notes

Cross-version relationships are drafted in data/curricula/crosswalks/csee-basic-mathematics-2005-to-mathematics-2023.json. Partial and 2005-only mappings remain reviewable.