Indices and Logarithms

1Summary 重點摘要

Learn index laws and basic logarithm operations, know conversion.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Know the index lawsIndex work rests on a few basic laws: multiplying powers with the same base adds the indices, a^m × a^n = a^(m+n); dividing powers with the same base subtracts the indices, a^m ÷ a^n = a^(m-n); and raising a power to a power multiplies the indices, (a^m)^n = a^(mn). Every law requires the bases to be identical before combining. A common mistake is multiplying the indices when multiplying powers: a² × a³ should be a⁵, not a⁶.
2. Step 2: Simplify index expressionsTo simplify an index expression, first write every part with the same base, then apply the laws in order to combine products and quotients into a single power. A negative index is handled by a^(-n) = 1/(a^n), and a zero index by a⁰ = 1. Apply one law at a time and state the reason, which reduces slips. A common mistake is confusing coefficients with indices: in 2a² × 3a³ the coefficients give 6 and the indices give a⁵.
3. Step 3: Understand what a logarithm meansA logarithm is the inverse of an index: if a^x = y, where a > 0 and a is not 1, then log_a y = x, read as the logarithm of y to base a. In other words, the logarithm answers the question of how many times the base must multiply itself to reach y. Once the definition is clear, every log calculation can be turned back into an index problem. A common mistake is treating log_a as a coefficient, for example writing log_a y as y ÷ a, which is wrong.
4. Step 4: Practise converting between formsIndex form and log form convert into each other: 2³ = 8 is equivalent to log₂8 = 3, and 10² = 100 is equivalent to log₁₀100 = 2. Hold on to the definition when practising: the logarithm is simply the index x itself. This conversion is the basic technique for solving log equations, translating any log form into index form before solving. A common mistake is swapping the base and the index, for example writing log₂8 = 3 as 3² = 8.
5. Step 5: Know the logarithm lawsLogarithms have their own laws: log_a (MN) = log_a M + log_a N, so a product splits into a sum of logs; log_a (M/N) = log_a M - log_a N, so a quotient splits into a difference; and log_a (M^n) = n log_a M, which brings the power out front. The laws need the same base and a positive number inside the log. A common mistake is assuming log(M + N) equals log M + log N; a sum cannot be split this way.
6. Step 6: Evaluate a logarithmTo evaluate a logarithm reliably, convert it back to index form: let log_a y = x, rewrite as a^x = y, then express y as a power of the base a, and that index is the answer. For example, to find log₃81, since 81 = 3⁴, the answer is 4. If the number inside the log is not an easy power of the base, simplify with the log laws first. A common mistake is ignoring that the log argument must be positive: log₂0 and log₂(-4) are meaningless.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Index laws: same base multiply add, divide subtract, power multiply
• Logarithm is opposite of index
• a^x = y ↔ log_a y = x
• log a + log b = log(ab)
• log a - log b = log(a/b)