Mathematics » Functions III » Exponential Functions

# Logarithms

## Logarithms

### Definition: Logarithm

If $$x = {b}^{y}$$, then $$y = {\log}_{b}(x)$$, where $$b>0$$, $$b \ne 1$$ and $$x>0$$.

Note that the brackets around the number $$(x)$$ are not compulsory, we use them to avoid confusion.

The logarithm of a number $$(x)$$ with a certain base $$(b)$$ is equal to the exponent $$(y)$$, the value to which that certain base must be raised to equal the number $$(x)$$.

For example, $$\log_{2}(8)$$ means the power of $$\text{2}$$ that will give $$\text{8}$$. Since $${2}^{3}=8$$, we see that $${\log}_{2}(8)=3$$. Therefore the exponential form is $${2}^{3}=8$$ and the logarithmic form is $${\log}_{2}{8}=3$$.

### Restrictions on the definition of logarithms

$\begin{array}{rll} \text{Restriction:}& & \text{Reason: } \\ & & \\ b > 0 & & \text{If } b \text{ is a negative number. then } b^{y} \text{ will oscillate between:} \\ & & \text{positive values if } y \text{ is even } \\ & & \text{negative values if } y \text{ is odd } \\ & & \\ b \ne 1 & & \text{Since } 1^{\text{(any value)}} = 1 \\ & & \\ x > 0 & & \text{Since } \text{(positive number)}^{\text{(any value)}} > 0\end{array}$

## Optional Investigation: Exponential and logarithmic form

Discuss the following statements and determine whether they are true or false:

1. $$p = a^{n}$$ is the inverse of $$p = \log_{a}{n}$$.
2. $$y = 2^{x}$$ is a one-to-one function, therefore $$y = \log_{2}{x}$$ is also a one-to-one function.
3. $$x = \log_{5}{y}$$ is the inverse of $$5^{x} = y$$.
4. $$k = b^{t}$$ is the same as $$t = \log_{b}{k}$$.

### To determine the inverse function of $$y=b^{x}$$:

$\begin{array}{rll} &(1) \quad \text{Interchange } x \text{ and } y: & x = b^{y} \\ &(2) \quad \text{Make } y \text{ the subject of the equation}: & y = \log_{b}{x} \end{array}$

Therefore, if we have the exponential function $$f(x) = b^{x}$$, then the inverse is the logarithmic function $$f^{-1}(x) = \log_{b}{x}$$.

The “common logarithm” has a base $$\text{10}$$ and can be written as $$\log_{10}{x} = \log{x}$$. In other words, the $$\log$$ symbol written without a base is interpreted as the logarithm to base $$\text{10}$$. For example, $$\log{\text{25}} = \log_{10}{\text{25}}$$.

## Example

### Question

Write the following exponential expressions in logarithmic form and express each in words:

1. $$5^{2} = 25$$
2. $$10^{-3} = \text{0.001}$$
3. $$p^{x} = q$$

### Determine the inverse of the given exponential expressions

Remember: $$m = a^{n}$$ is the same as $$n = \log_{a}{m}$$.

1. $$2 = \log_{5}{25}$$
2. $$-3 = \log_{10}{(\text{0.001})}$$
3. $$x = \log_{p}{q}$$

### Express in words

1. $$\text{2}$$ is the power to which $$\text{5}$$ must be raised to give the number $$\text{25}$$.
2. $$-\text{3}$$ is the power to which $$\text{10}$$ must be raised to give the decimal number $$\text{0.001}$$.
3. $$x$$ is the power to which $$p$$ must be raised to give $$q$$.

## Example

### Question

Write the following logarithmic expressions in exponential form:

1. $$\log_{2}{128} = 7$$
2. $$-2 = \log_{3}{( \cfrac{1}{9} )}$$
3. $$z = \log_{w}{k}$$

### Determine the inverse of the given logarithmic expressions

For $$n = \log_{a}{m}$$, we can write $$m = a^{n}$$.

1. $$2^{7} = \text{128}$$
2. $$3^{-2} = \cfrac{1}{9}$$
3. $$w^{z} = k$$

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