Mathematics » Functions III » Exponential Functions

# Laws of Logarithms

## Laws of Logarithms

In earlier grades, we used the following exponential laws for working with exponents:

• $${a}^{m} \times {a}^{n}={a}^{m+n}$$
• $$\cfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}$$
• $${(ab)}^{n}={a}^{n}{b}^{n}$$
• $${\left(\cfrac{a}{b}\right)}^{n}=\cfrac{{a}^{n}}{{b}^{n}}$$
• $${({a}^{m})}^{n}={a}^{mn}$$

where $$a > 0$$, $$b > 0$$ and $$m, n \in ℤ$$.

The logarithmic laws are based on the exponential laws and make working with logarithms much easier.

### Logarithmic laws:

• $$\log_{a}{x^{b}} = b \log_{a}{x} \qquad (x > 0)$$
• $$\log_{a}{x} = \cfrac{\log_{b}{x}}{\log_{b}{a}} \qquad (b > 0 \text{ and } b \ne 1)$$
• $$\log_{a}{xy} = \log_{a}{x} + \log_{a}{y} \qquad (x > 0 \text{ and } y > 0)$$
• $$\log_{a}{\cfrac{x}{y}} = \log_{a}{x} – \log_{a}{y} \qquad (x > 0 \text{ and } y > 0)$$

The last two logarithmic laws in the list above are not covered in this section. They are discussed at the end of the tutorial and are included for enrichment only.

### Logarithmic law:

$\log_{a}{x^{b}} = b \log_{a}{x} \qquad (x > 0 )$

\begin{align*} \text{Let } {\log}_{a}{x} &= m \ldots (1)\qquad (x > 0 ) \\ \therefore x &= a^{m} \\ \therefore (x)^{b} &= ( a^{m} )^{b} \\ \therefore x^{b} &= a^{bm} \\ \text{Change to logarithmic form: }\log_{a}(x^{b}) &= bm \\ \text{And subst}: \quad m &= {\log}_{a}{x} \\ \therefore {\log}_{a}{x^{b}} &= b{\log}_{a}{x} \end{align*}

In words: the logarithm of a number which is raised to a power is equal to the value of the power multiplied by the logarithm of the number.

## Example

### Question

Determine the value of $$\log_{3}{{27}^{4}}$$.

### Use the logarithmic law to simplify the expression

\begin{align*} \log_{3}{{27}^{4}} &= 4 \log_{3}{{27}}\\ &= 4\log_{3}{{3}^{3}} \\ &= (4 \times 3) \log_{3}{3}\\ &= 12 (1) \\ &= 12 \end{align*}

$$\log_{3}{{27}^{4}} = 12$$

### Special case:

$\log_{a}{\sqrt[b]{x}} = \cfrac{\log_{a}{x}}{b} \qquad (x > 0 \text{ and } b > 0 )$

The following is a special case of the logarithmic law $$\log_{a}{x^{b}} = b \log_{a}{x}$$:

\begin{align*} \log_{a}{\sqrt[b]{x}} &= \log_{a}{{x}^{\frac{1}{b}}} \\ &= \cfrac{1}{b} \log_{a}{x} \\ &= \cfrac{\log_{a}{x}}{b} \end{align*}

### Logarithmic law:

$${\log}_{a}x=\cfrac{{\log}_{b}{x}}{{\log}_{b}{a}} \qquad (b > 0 \text{ and } b \ne 1)$$

It is often necessary or convenient to convert a logarithm from one base to another base. This is referred to as a change of base.

\begin{align*} \text{Let } \quad \log_{a}{x} &= m \\ \therefore x &= a^{m} \\ \text{Consider the fraction: } \quad & \cfrac{\log_{b}{x}}{\log_{b}{a}} \\ \text{Substitute } x = a^{m}: \quad \cfrac{\log_{b}{x}}{\log_{b}{a}} &= \cfrac{\log_{b}{a^{m}}}{\log_{b}{a}} \\ &= m ( \cfrac{\log_{b}{a}}{\log_{b}{a}} ) \\ &= m (1) \\ \therefore \cfrac{\log_{b}{x}}{\log_{b}{a}}&= \log_{a}{x} \end{align*}

### Special applications:

\begin{align*} (1) \qquad \log_{a}{x} &= \cfrac{\log_{x}{x}}{\log_{x}{a}} \\ \therefore \log_{a}{x} &= \cfrac{1}{\log_{x}{a}} \\ & \\ & \\ (2) \qquad \log_{a}{\cfrac{1}{x}} &= \log_{a}{x^{-1}} \\ \therefore \log_{a}{\cfrac{1}{x}} &= – \log_{a}{x} \end{align*}

## Example

### Question

Show: $$\log_{2}{8} = \cfrac{\log{8}}{\log{2}}$$

### Simplify the right-hand side of the equation

\begin{align*} \text{RHS } &= \cfrac{\log{8}}{\log{2}} \\ &= \cfrac{\log{2^{3}}}{\log{2}} \\ &= 3 ( \cfrac{\log{2}}{\log{2}} ) \\ &= 3(1) \\ &= 3 \end{align*}

### Simplify the left-hand side of the equation

\begin{align*} \text{LHS } &= \log_{2}{8} \\ &= \log_{2}{2^{3}} \\ &= 3 \log_{2}{2} \\ &= 3(1) \\ &= 3 \end{align*}

We have shown that $$\log_{2}{8} = \dfrac{\log{8}}{\log{2}} = 3$$.

## Example

### Question

If $$a = \log{2}$$ and $$b = \log{3}$$, express the following in terms of $$a$$ and $$b$$:

1. $$\log_{3}{2}$$
2. $$\log_{2}{\cfrac{10}{3}}$$

### Use a change of base to simplify the expressions

1. \begin{align*} \log_{3}{2} &= \cfrac{\log{2}}{\log{3}} \\ &= \cfrac{a}{b} \end{align*}
2. \begin{align*} \log_{2}{\cfrac{10}{3}} &= \cfrac{\log{\cfrac{10}{3}}}{\log{2}} \\ &= \cfrac{\log{10} – \log{3}}{\log{2}} \\ &= \cfrac{1- b}{a} \end{align*}

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