Required Prior Knowledge

Questions

Calculate

a) $6a^{3}b^{2}\times 2a^{8}b^{3}$

b) $10x^{9}y\div 5x^{3}y^{7}$

c) $5x^{\frac{1}{2}}\left(x^{-\frac{1}{2}}-2x^{\frac{3}{2}}\right)$

d) $\sqrt{9^{16x^{2}}}$

e) $\frac{16^{x+1}+20\times 4^{2x}}{2^{x-3}\times 8^{x+2}}$

State the Laws of Indices.

Solutions

Get Ready

Question

Read through this proof. At each stage think carefully about what is being said and what is shows.

Let

$x=\log_{b}m$

This implies that

$b^{x}=m$

Similarly

$y=\log_{b}n\Rightarrow b^{y}=n$

Hence

$\begin{align}\log_{b}mn &=\log_{b}b^{x}b^{y}\\&=\log_{b}b^{x+y}\\&=x+y\\&=\log_{b}m +\log_{b}n\end{align}$

Thus

$\log_{b}m +\log_{b}n=\log_{b}mn$

Explanation

Notes

There are four Laws of Logarithms which derive from the Laws of Indices.

The first of these is proven in the Get Ready.

Think about what the result for the other three might be and then check.

a) $\log_{b}m+\log_{b}n=\log_{b}mn$

b) $\log_{b}m-\log_{b}n=$

c) $k\log_{b}m=$

d) $-\log_{b}m=$

a) $\log_{b}m+\log_{b}n=\log_{b}mn$
b) $\log_{b}m-\log_{b}n=\log_{b}\frac{m}{n}$
c) $k\log_{b}m=\log_{b}m^{k}$
d) $-\log_{b}m=\log_{b}\frac{1}{m}$

Use the proof in the Get Ready to construct similar proofs for the other three Laws of Logarithms.

Examples and Your Turns

Example

Express $\log_{a}5+2\log_{a}7-\log_{a}35$ as a single logarithm.

Your Turn

Express $\log_{a}p+2\log_{a}q-3\log_{a}r$ as a single logarithm.

Your Turn

Express $1-\log_{a}ab$ as a single logarithm.

Your Turn

Evaluate $2\left(\log 5 + \log 2\right)-1$

Your Turn

Simplify $2\log_{5}4+3$

Your Turn

Simplify $\frac{\log 8}{\log 4}$

Your Turn

In each of these groups of three expressions, two are equal. Determine the odd one out, and write a fourth expression to match the value of the odd one out.

a) $\log_{3}27,\quad \log_{4}16,\quad \log_{5}125$

b) $\log_{4}0.25,\quad \log_{5}0.2,\quad \log_{10}0.01$

c) $\log_{2}x^{4} +\log_{2}x^{3},\quad \log_{2}x^{7},\quad \log_{2}x^{2} +\log_{2}x^{6}$

d) $-2\log_{5}x,\quad \log_{5}x^{3}-\log_{5}x^{5},\quad \log_{5}\sqrt{x}$

$\log_{3}27=3\\ \log_{4}16=2\\ \log_{5}125=3$
Hence the odd one out is $\log_{4}16$ .
Possible matching values could be $\log_{3}9,\log_{2}4,\log_{x}x^{2},...$
$\log_{4}0.25=-1\\ \log_{5}0.2=-1\\ \log_{10}0.01=-2$
Hence the odd one out is $\log_{10}0.01$ .
Possible matching values are $\log_{2}0.25,\log_{0.1}100,\log_{x}\frac{1}{x^{2}},...$
$\log_{2}x^{4} +\log_{2}x^{3}=\log_{2}x^{12}\\ \log_{2}x^{7}\\ \log_{2}x^{2} +\log_{2}x^{6}=\log_{2}x^{12}$
Hence the odd one out is $\log_{2}x^{7}$ .
Possible matching values could be $\log_{2}x +\log_{2}x^{6},\log_{2}x^{12}-\log_{2}x^{5},...$
$-2\log_{5}x=\log_{5}x^{-2}\\ \log_{5}x^{3}-\log_{5}x^{5}=\log_{5}x^{-2}\\ \log_{5}\sqrt{x}=\log_{5}x^{\frac{1}{2}}$
Hence the odd one out is $\log_{5}\sqrt{x}$ .
A possible matching value is $\frac{1}{2}\log_{5}x$

Key Facts

Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.

Taking it Deeper

Conceptual Questions to Consider

Why can we not simplify $\log\left(a+b\right)$ ?

Common Mistakes / Misconceptions

Misremembering the laws of logarithms e.g. $\log\left(a+b\right)\ne\log\left(a\right)+\log\left(b\right)$

or \(\log\left(\frac{a}{b}\right)\ne\frac{\log\left(a\right)}{\log\left(b\right)}

Thinking you can cancel logarithms like $\frac{\log 12}{\log 8}$ down to $\frac{3}{2}$ .

Connecting This to Other Skills

Laws of logarithms are used to simplify expressions involving logarithms, and solve equations. These will be important in Geometric Sequences (1.5) and Equations involving exponentials (1.12).

Self-Reflection

What was the most challenging part of this skill for you?

What are you still unsure about that you need to review?

1-2 Laws of Logarithms

Required Prior Knowledge

Get Ready

Notes

Examples and Your Turns

Key Facts

Taking it Deeper