Learning Objectives
In this section, you will:
4.2.1 – Use the product rule for logarithms.
4.2.2 – Use the quotient rule for logarithms.
4.2.3 – Use the power rule for logarithms.
4.2.4 – Expand logarithmic expressions.
In chemistry, pH is used as a measure of the acidity or alkalinity of a substance. The pH scale runs from 0 to 14. Substances with a pH less than 7 are considered acidic, and substances with a pH greater than 7 are said to be alkaline. Our bodies, for instance, must maintain a pH close to 7.35 in order for enzymes to work properly. To get a feel for what is acidic and what is alkaline, consider the following pH levels of some common substances:
- Battery acid: 0.8
- Stomach acid: 2.7
- Orange juice: 3.3
- Pure water: 7 (at 25° C)
- Human blood: 7.35
- Fresh coconut: 7.8
- Sodium hydroxide (lye): 14
To determine whether a solution is acidic or alkaline, we find its pH, which is a measure of the number of active positive hydrogen ions in the solution. The pH is defined by the following formula, where [latex]a[/latex] is the concentration of hydrogen ion in the solution
The equivalence of [latex]-\mathrm{log}\left(\left[{H}^{+}\right]\right)[/latex] and [latex]\mathrm{log}\left(\frac{1}{\left[{H}^{+}\right]}\right)[/latex] is one of the logarithm properties we will examine in this section.
4.2.1 – Using the Product Rule for Logarithms
Recall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here. First, the following properties are easy to prove.
[latex]$$\begin{array}{l}{\mathrm{log}}_{b}1=0\\ {\mathrm{log}}_{b}b=1\end{array} $$[/latex]
For example, [latex]{\mathrm{log}}_{5}1=0[/latex] since [latex]{5}^{0}=1.[/latex] And [latex]{\mathrm{log}}_{5}5=1[/latex] since [latex]{5}^{1}=5.[/latex]
Next, we have the inverse property.
[latex]$$ \begin{array}{l}\hfill \\ {\mathrm{log}}_{b}\left({b}^{x}\right)=x\hfill \\ \text{ }{b}^{{\mathrm{log}}_{b}x}=x,x>0\hfill \end{array}$$[/latex]
For example, to evaluate [latex]\mathrm{log}\left(100\right),[/latex] we can rewrite the logarithm as [latex]{\mathrm{log}}_{10}\left({10}^{2}\right),[/latex] and then apply the inverse property [latex]{\mathrm{log}}_{b}\left({b}^{x}\right)=x[/latex] to get [latex]{\mathrm{log}}_{10}\left({10}^{2}\right)=2.[/latex]
To evaluate [latex]{e}^{\mathrm{ln}\left(7\right)},[/latex] we can rewrite the logarithm as [latex]{e}^{{\mathrm{log}}_{e}7},[/latex] and then apply the inverse property [latex]{b}^{{\mathrm{log}}_{b}x}=x[/latex] to get [latex]{e}^{{\mathrm{log}}_{e}7}=7.[/latex]
Finally, we have the one-to-one property.
[latex]{\mathrm{log}}_{b}M={\mathrm{log}}_{b}N\,\text{ if and only if}\,\text{ }M=N[/latex]
We can use the one-to-one property to solve the equation [latex]{\mathrm{log}}_{3}\left(3x\right)={\mathrm{log}}_{3}\left(2x+5\right)[/latex] for [latex]x.[/latex] Since the bases are the same, we can apply the one-to-one property