Figure 1. The pH of hydrochloric acid is tested with litmus paper. (credit: David Berardan)

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 a 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.

Use 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.

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.

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.

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 x. Since the bases are the same, we can apply the one-to-one property by setting the arguments equal and solving for x:

But what about the equation [latex]{\mathrm{log}}_{3}\left(3x\right)+{\mathrm{log}}_{3}\left(2x+5\right)=2[/latex]? The one-to-one property does not help us in this instance. Before we can solve an equation like this, we need a method for combining terms on the left side of the equation.

Recall that we use the product rule of exponents to combine the product of exponents by adding: [latex]{x}^{a}{x}^{b}={x}^{a+b}[/latex]. We have a similar property for logarithms, called the product rule for logarithms, which says that the logarithm of a product is equal to a sum of logarithms. Because logs are exponents, and we multiply like bases, we can add the exponents. We will use the inverse property to derive the product rule below.

Given any real number x and positive real numbers M, N, and b, where [latex]b\ne 1[/latex], we will show

Let [latex]m={\mathrm{log}}_{b}M[/latex] and [latex]n={\mathrm{log}}_{b}N[/latex]. In exponential form, these equations are [latex]{b}^{m}=M[/latex] and [latex]{b}^{n}=N[/latex]. It follows that

Note that repeated applications of the product rule for logarithms allow us to simplify the logarithm of the product of any number of factors. For example, consider [latex]{\mathrm{log}}_{b}\left(wxyz\right)[/latex]. Using the product rule for logarithms, we can rewrite this logarithm of a product as the sum of logarithms of its factors:

Use the quotient and power rules for logarithms

We can use the rules of logarithms we just learned to condense sums, differences, and products with the same base as a single logarithm. It is important to remember that the logarithms must have the same base to be combined. We will learn later how to change the base of any logarithm before condensing.

 Use the change-of-base formula for logarithms

Key Equations

Section 3.3 Homework Exercises

1. How does the power rule for logarithms help when solving logarithms with the form [latex]{\mathrm{log}}_{b}\left(\sqrt[n]{x}\right)[/latex]?

2. What does the change-of-base formula do? Why is it useful when using a calculator?

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

3. [latex]{\mathrm{log}}_{b}\left(7x\cdot 2y\right)[/latex]

4. [latex]\mathrm{ln}\left(3ab\cdot 5c\right)[/latex]

5. [latex]{\mathrm{log}}_{b}\left(\frac{13}{17}\right)[/latex]

6. [latex]{\mathrm{log}}_{4}\left(\frac{\text{ }\frac{x}{z}\text{ }}{w}\right)[/latex]

7. [latex]\mathrm{ln}\left(\frac{1}{{4}^{k}}\right)[/latex]

8. [latex]{\mathrm{log}}_{2}\left({y}^{x}\right)[/latex]

For the following exercises, condense to a single logarithm if possible.

9. [latex]\mathrm{ln}\left(7\right)+\mathrm{ln}\left(x\right)+\mathrm{ln}\left(y\right)[/latex]

10. [latex]{\mathrm{log}}_{3}\left(2\right)+{\mathrm{log}}_{3}\left(a\right)+{\mathrm{log}}_{3}\left(11\right)+{\mathrm{log}}_{3}\left(b\right)[/latex]

11. [latex]{\mathrm{log}}_{b}\left(28\right)-{\mathrm{log}}_{b}\left(7\right)[/latex]

12. [latex]\mathrm{ln}\left(a\right)-\mathrm{ln}\left(d\right)-\mathrm{ln}\left(c\right)[/latex]

13. [latex]-{\mathrm{log}}_{b}\left(\frac{1}{7}\right)[/latex]

14. [latex]\frac{1}{3}\mathrm{ln}\left(8\right)[/latex]

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

15. [latex]\mathrm{log}\left(\frac{{x}^{15}{y}^{13}}{{z}^{19}}\right)[/latex]

16. [latex]\mathrm{ln}\left(\frac{{a}^{-2}}{{b}^{-4}{c}^{5}}\right)[/latex]

17. [latex]\mathrm{log}\left(\sqrt{{x}^{3}{y}^{-4}}\right)[/latex]

18. [latex]\mathrm{ln}\left(y\sqrt{\frac{y}{1-y}}\right)[/latex]

19. [latex]\mathrm{log}\left({x}^{2}{y}^{3}\sqrt[3]{{x}^{2}{y}^{5}}\right)[/latex]

For the following exercises, condense each expression to a single logarithm using the properties of logarithms.

20. [latex]\mathrm{log}\left(2{x}^{4}\right)+\mathrm{log}\left(3{x}^{5}\right)[/latex]

21. [latex]\mathrm{ln}\left(6{x}^{9}\right)-\mathrm{ln}\left(3{x}^{2}\right)[/latex]

22. [latex]2\mathrm{log}\left(x\right)+3\mathrm{log}\left(x+1\right)[/latex]

23. [latex]\mathrm{log}\left(x\right)-\frac{1}{2}\mathrm{log}\left(y\right)+3\mathrm{log}\left(z\right)[/latex]

24. [latex]4{\mathrm{log}}_{7}\left(c\right)+\frac{{\mathrm{log}}_{7}\left(a\right)}{3}+\frac{{\mathrm{log}}_{7}\left(b\right)}{3}[/latex]

For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.

25. [latex]{\mathrm{log}}_{7}\left(15\right)[/latex] to base e

26. [latex]{\mathrm{log}}_{14}\left(55.875\right)[/latex] to base 10

For the following exercises, suppose [latex]{\mathrm{log}}_{5}\left(6\right)=a[/latex] and [latex]{\mathrm{log}}_{5}\left(11\right)=b[/latex]. Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of a and b. Show the steps for solving.

27. [latex]{\mathrm{log}}_{11}\left(5\right)[/latex]

28. [latex]{\mathrm{log}}_{6}\left(55\right)[/latex]

29. [latex]{\mathrm{log}}_{11}\left(\frac{6}{11}\right)[/latex]

For the following exercises, use properties of logarithms to evaluate without using a calculator.

30. [latex]{\mathrm{log}}_{3}\left(\frac{1}{9}\right)-3{\mathrm{log}}_{3}\left(3\right)[/latex]

31. [latex]6{\mathrm{log}}_{8}\left(2\right)+\frac{{\mathrm{log}}_{8}\left(64\right)}{3{\mathrm{log}}_{8}\left(4\right)}[/latex]

32. [latex]2{\mathrm{log}}_{9}\left(3\right)-4{\mathrm{log}}_{9}\left(3\right)+{\mathrm{log}}_{9}\left(\frac{1}{729}\right)[/latex]

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.

33. [latex]{\mathrm{log}}_{3}\left(22\right)[/latex]

34. [latex]{\mathrm{log}}_{8}\left(65\right)[/latex]

35. [latex]{\mathrm{log}}_{6}\left(5.38\right)[/latex]

36. [latex]{\mathrm{log}}_{4}\left(\frac{15}{2}\right)[/latex]

37. [latex]{\mathrm{log}}_{\frac{1}{2}}\left(4.7\right)[/latex]

38. Use the product rule for logarithms to find all x values such that [latex]{\mathrm{log}}_{12}\left(2x+6\right)+{\mathrm{log}}_{12}\left(x+2\right)=2[/latex]. Show the steps for solving.

39. Use the quotient rule for logarithms to find all x values such that [latex]{\mathrm{log}}_{6}\left(x+2\right)-{\mathrm{log}}_{6}\left(x - 3\right)=1[/latex]. Show the steps for solving.

40. Can the power property of logarithms be derived from the power property of exponents using the equation [latex]{b}^{x}=m?[/latex] If not, explain why. If so, show the derivation.

41. Prove that [latex]{\mathrm{log}}_{b}\left(n\right)=\frac{1}{{\mathrm{log}}_{n}\left(b\right)}[/latex] for any positive integers b > 1 and n > 1.

42. Does [latex]{\mathrm{log}}_{81}\left(2401\right)={\mathrm{log}}_{3}\left(7\right)[/latex]? Verify the claim algebraically.