TheIBAxolotl
Powers as repeated multiplication
\(a^n\) means \(n\) copies of \(a\) multiplied together. From that one idea, the index laws follow by counting factors:
- \(a^m \times a^n = a^{m+n}\) (you just count all the factors),
- \(a^m \div a^n = a^{m-n}\) (factors cancel),
- \((a^m)^n = a^{mn}\),
- \((ab)^n = a^n b^n\).
To keep the division law working when \(m = n\) we need \(a^0 = 1\), and when \(m < n\) we need \(a^{-n} = \frac{1}{a^n}\). So a negative power means "reciprocal", not "negative number": \(2^{-3} = \frac{1}{8}\).
Worked example: simplifying
Simplify \(\dfrac{(2x^3)^2}{4x^{-1}}\).
The numerator is \(4x^6\). Dividing, \(\frac{4x^6}{4x^{-1}} = x^{6-(-1)} = x^7\).
What a logarithm is
A logarithm is the inverse of a power. The statement
\[\log_a b = x \iff a^x = b \qquad (a > 0,\ a \ne 1,\ b > 0)\]
means "\(x\) is the power you raise \(a\) to in order to get \(b\)". So \(\log_2 32 = 5\) because \(2^5 = 32\), and \(\log_3 \frac{1}{9} = -2\) because \(3^{-2} = \frac{1}{9}\).
Two bases have their own buttons:
- \(\log x\) means \(\log_{10} x\) (common log),
- \(\ln x\) means \(\log_e x\) (natural log), where \(e \approx 2.718\).
Because a positive base raised to any power is positive, you cannot take the log of zero or a negative number.
Switching forms
Most log questions are solved by rewriting in exponential form, or the other way round.
- \(\log_5 x = 3 \Rightarrow x = 5^3 = 125\).
- \(10^x = 50 \Rightarrow x = \log 50 \approx 1.70\).
- \(e^{t} = 4 \Rightarrow t = \ln 4 \approx 1.39\).
Two facts fall straight out of the definition: \(\log_a a^x = x\) and \(a^{\log_a x} = x\). They say that powers and logs undo each other, just as squaring and square-rooting do for positive numbers.
Using technology
On calculator papers, evaluate logs directly; for other bases most GDCs have a \(\log_a\) template. On non-calculator papers, the questions are chosen so that the answer is a neat power: look for it by writing the number as a power of the base.
Common mistakes
- Treating \(a^{-n}\) as \(-a^n\).
- Adding powers when bases are different, as in \(2^3 \times 3^2 \ne 6^5\).
- Writing \((a + b)^2 = a^2 + b^2\); index laws apply to products, not sums.
- Taking the log of a negative number or zero.
- Confusing \(\log\) (base 10) with \(\ln\) (base \(e\)).
Exam tips
- Rewrite roots and reciprocals as powers before simplifying.
- For exact log values, express both numbers as powers of the same base.
- When converting, say the sentence "base to the answer equals the number" to yourself.