## 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:

- \(p = a^{n}\) is the inverse of \(p = \log_{a}{n}\).
- \(y = 2^{x}\) is a one-to-one function, therefore \(y = \log_{2}{x}\) is also a one-to-one function.
- \(x = \log_{5}{y}\) is the inverse of \(5^{x} = y\).
- \(k = b^{t}\) 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:

- \(5^{2} = 25\)
- \(10^{-3} = \text{0.001}\)
- \(p^{x} = q\)

### Determine the inverse of the given exponential expressions

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

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

### Express in words

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

## Example

### Question

Write the following logarithmic expressions in exponential form:

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

### Determine the inverse of the given logarithmic expressions

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

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