1.5 Laws of exponents and introduction to logarithms (IB Maths: Analysis and Approaches SL): Revision notes

Section 1

What are the laws of exponents?

For a non-zero base $a$ and integers $m$ and $n$ :

$a^{m}\times a^{n} = a^{m+n}$
$a^{m}\div a^{n} = a^{m-n}$
$(a^{m})^{n} = a^{mn}$
$(ab)^{n} = a^{n}b^{n}$
$a^{0} = 1$ and $a^{-n} = \dfrac{1}{a^{n}}$

Examples: $5^{3}\times5^{-6} = 5^{-3}$ , $(2^{3})^{-4} = 2^{-12}$ , $(2x)^{4} = 16x^{4}$ , $2x^{-3} = \dfrac{2}{x^{3}}$ .

Key termsexponentbasenegative exponent

Common mistake

$2x^{-3}$ is $\frac{2}{x^{3}}$ , not $\frac{1}{2x^{3}}$ — the exponent applies only to $x$ . But $(2x)^{-3} = \frac{1}{8x^{3}}$ .

Common mistake

$(2x^{3})^{-2}$ : the power applies to the 2 as well, giving $\frac{1}{4x^{6}}$ , not $\frac{1}{2x^{6}}$ .

Section 2

Simplifying expressions with integer exponents

Deal with coefficients and each letter separately. For example $\frac{6x^{-2}}{3x^{-5}} = \frac{6}{3}x^{-2-(-5)} = 2x^{3}.$ Write the final answer in the form requested — either with negative exponents ( $\frac{1}{2}x^{-3}$ ) or as a fraction with positive exponents ( $\frac{1}{2x^{3}}$ ).

Watch subtraction of negative exponents: $-2 - (-5) = 3$ .

Key termscoefficient

Exam tip

Bracket negative exponents when subtracting them: $x^{-2}\div x^{-5} = x^{(-2)-(-5)}$ .

Section 3

What is a logarithm?

A logarithm answers the question "what power?". For $a > 0$ , $a \ne 1$ and $b > 0$ : $a^{x} = b \Leftrightarrow \log_{a}b = x.$ So $\log_{10}1000 = 3$ because $10^{3} = 1000$ , and $\log_{10}0.01 = -2$ because $10^{-2} = 0.01$ .

You can only take the logarithm of a positive number: $\log_{10}(-5)$ and $\log_{10}0$ are undefined, because no power of 10 is zero or negative.

Key termslogarithm

Example

pH $= -\log_{10}c$ . If pH $= 2.5$ then $\log_{10}c = -2.5$ , so $c = 10^{-2.5}$ .

Section 4

Logarithms base 10 and base e

At SL (in this subtopic) you work with two bases:

common logarithm $\log_{10}x$ (often written $\log x$ on calculators);
natural logarithm $\ln x = \log_{e}x$ , where $e \approx 2.718$ .

Using the equivalence: $e^{x} = b \Leftrightarrow x = \ln b$ and $10^{x} = b \Leftrightarrow x = \log_{10}b$ . For example $e^{0.3t} = 8 \Rightarrow 0.3t = \ln8 \Rightarrow t = 6.93$ .

Evaluate logarithms numerically with technology, e.g. $-\log_{10}(3.2\times10^{-5}) = 4.49$ .

Key termsnatural logarithmcommon logarithme

Common mistake

Typing $\ln8/0.3$ correctly but then writing $\ln(8/0.3)$ . Keep the brackets clear: $t = \frac{\ln8}{0.3}$ .

Section 5

Logarithmic scales in context

Many real scales use logarithms so that huge ranges fit onto small numbers: pH, decibels, the Richter scale.

On the pH scale, a drop of 1 means 10 times the hydrogen ion concentration.
On the decibel scale $L = 10\log_{10}\left(\frac{I}{I_0}\right)$ , an increase of 10 dB means 10 times the intensity; 70 dB to 110 dB is $10^{4}$ times the intensity.

To reverse a logarithmic formula, isolate the log, then rewrite in exponential form: $\log_{10}\left(\frac{I}{I_0}\right) = 11 \Rightarrow \frac{I}{I_0} = 10^{11}$ .

Key termslogarithmic scale

Exam tip

When a context gives a maximum (like 100 dB), round your final whole-number answer down so the limit is not exceeded.

Must know

$a^{m}a^{n} = a^{m+n}$ , $\frac{a^{m}}{a^{n}} = a^{m-n}$ , $(a^{m})^{n} = a^{mn}$ , $a^{-n} = \frac{1}{a^{n}}$ , $a^{0} = 1$ .
A power outside a bracket applies to every factor inside: $(2x)^{4} = 16x^{4}$ .
$a^{x} = b \Leftrightarrow \log_{a}b = x$ , with $b > 0$ .
$\log_{e}x = \ln x$ ; use technology to evaluate logarithms.
To solve $e^{kx} = b$ , write $kx = \ln b$ .

That's the notes covered.

Carry on to the next subtopic.