Surds and Logarithmic Functions

Chapter 1 student notes

1. Core language: indices, surds and exact values

Exact arithmetic is a frequent gateway skill: it reappears inside binomial approximation, coordinate geometry and trigonometry.

Index laws
\[a^m a^n=a^{m+n},\quad \frac{a^m}{a^n}=a^{m-n},\quad (a^m)^n=a^{mn}\]
Fractional & negative powers
\[a^{1/n}=\sqrt[n]{a},\qquad a^{-n}=\frac1{a^n}\]
Surd products
\[\sqrt a\sqrt b=\sqrt{ab},\qquad \frac{\sqrt a}{\sqrt b}=\sqrt{\frac ab}\]
Conjugates
\[(p+q\sqrt r)(p-q\sqrt r)=p^2-q^2r\]

A surd is an exact irrational root that is deliberately left in radical form. Simplify by extracting perfect-square factors before adding like surds. Only terms with the same radical part combine.

Worked example — simplify and rationalise

\[\sqrt{72}-2\sqrt 8=6\sqrt2-4\sqrt2=2\sqrt2.\]

For \(\frac{5}{3-\sqrt2}\), multiply numerator and denominator by the conjugate:

\[\frac{5}{3-\sqrt2}\cdot\frac{3+\sqrt2}{3+\sqrt2}=\frac{5(3+\sqrt2)}{9-2}=\frac{15+5\sqrt2}{7}.\]
Common mistake. Do not replace an exact surd by a decimal unless the question explicitly asks for a numerical approximation.

2. Logarithms: meaning and laws

Logarithms convert multiplication into addition and powers into coefficients; domain checking is part of the method, not an optional final step.

Definition
\[\log_a x=y\iff a^y=x\qquad (a>0,\ a e1,\ x>0)\]
Product / quotient
\[\log_a(MN)=\log_aM+\log_aN,\quad \log_a\!\left(\frac MN\right)=\log_aM-\log_aN\]
Power rule
\[\log_a(M^k)=k\log_aM\]
Change of base
\[\log_a x=\frac{\log_bx}{\log_ba},\qquad \log_ab=\frac1{\log_ba}\]

Worked example — solve a logarithmic equation

Solve \(\log_2(x-1)+\log_2(x+2)=3\). \[\log_2[(x-1)(x+2)]=3\Rightarrow (x-1)(x+2)=8.\] \[x^2+x-10=0\Rightarrow x=\frac{-1\pm\sqrt{41}}2.\]

The domain requires \(x-1>0\), so only \(\displaystyle x=\frac{-1+\sqrt{41}}2\) is valid.

Exam tip. Write domain restrictions before or immediately after solving. Log arguments must be positive; extraneous roots are a common source of lost marks.
Common mistake. There is no law \(\log(a+b)=\log a+\log b\). Log laws act on products, quotients and powers, not sums.

3. Exponential and logarithmic graphs

Students should be able to recognise, sketch, transform and use these graphs to estimate solutions.

xy y = 2ˣy = log₂xy = xThe two curves are exact inverses, reflected in y = x.
Exponential
\[y=a^x:\quad (0,1),\quad y>0,\quad y=0\text{ is a horizontal asymptote}\]
Logarithm
\[y=\log_a x:\quad (1,0),\quad x>0,\quad x=0\text{ is a vertical asymptote}\]
Inverse relation
\[y=a^x\ \text{and}\ y=\log_a x\ \text{are reflections in }y=x\]
Transformation
\[y=Aa^{k(x-h)}+c\Rightarrow\text{shift, stretch/reflection and new asymptote }y=c\]
Exam tip. For a sketch, mark intercepts and asymptotes first, then the direction/shape. A correct shape without the defining features may not earn full marks.

4. Solving exponential and logarithmic equations

Choose the algebraic route that exposes structure; calculator logs are usually the final step, not the first.

  1. If possible, rewrite both sides with a common base.
  2. Otherwise take logarithms and use the power rule to bring unknown exponents down.
  3. For equations containing several logarithms, combine them before converting to exponential form.
  4. Solve the resulting algebraic equation, then check every log argument is positive.

Worked example — unknown in an exponent

\[3^{2x-1}=7\Rightarrow (2x-1)\ln3=\ln7\] \[x=\frac12\left(1+\frac{\ln7}{\ln3}\right).\]

Worked example — parameter form

If \(\log_a2=p\) and \(\log_a3=q\), then \[\log_a\left(\frac{9\sqrt2}{4}\right)=2q+\frac12p-2p=2q-\frac32p.\]

5. Chapter mastery checklist

Mastery check. If all six boxes are comfortable, you have the method set needed for the Surds and Logarithmic Functions topic-question bank.