Condense logarithmic expressions
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.
How To: Given a sum, difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm.
- Apply the power property first. Identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power.
- Next apply the product property. Rewrite sums of logarithms as the logarithm of a product.
- Apply the quotient property last. Rewrite differences of logarithms as the logarithm of a quotient.
Example 9: Using the Product and Quotient Rules to Combine Logarithms
Write [latex]{\mathrm{log}}_{3}\left(5\right)+{\mathrm{log}}_{3}\left(8\right)-{\mathrm{log}}_{3}\left(2\right)[/latex] as a single logarithm.
Solution
Using the product and quotient rules
This reduces our original expression to
Then, using the quotient rule
Try It 9
Condense [latex]\mathrm{log}3-\mathrm{log}4+\mathrm{log}5-\mathrm{log}6[/latex].
SolutionExample 10: Condensing Complex Logarithmic Expressions
Condense [latex]{\mathrm{log}}_{2}\left({x}^{2}\right)+\frac{1}{2}{\mathrm{log}}_{2}\left(x - 1\right)-3{\mathrm{log}}_{2}\left({\left(x+3\right)}^{2}\right)[/latex].
Solution
We apply the power rule first:
Next we apply the product rule to the sum:
Finally, we apply the quotient rule to the difference:
Example 11: Rewriting as a Single Logarithm
Rewrite [latex]2\mathrm{log}x - 4\mathrm{log}\left(x+5\right)+\frac{1}{x}\mathrm{log}\left(3x+5\right)[/latex] as a single logarithm.
Solution
We apply the power rule first:
Next we apply the product rule to the sum:
Finally, we apply the quotient rule to the difference:
Try It 10
Rewrite [latex]\mathrm{log}\left(5\right)+0.5\mathrm{log}\left(x\right)-\mathrm{log}\left(7x - 1\right)+3\mathrm{log}\left(x - 1\right)[/latex] as a single logarithm.
SolutionTry It 11
Condense [latex]4\left(3\mathrm{log}\left(x\right)+\mathrm{log}\left(x+5\right)-\mathrm{log}\left(2x+3\right)\right)[/latex].
SolutionExample 12: Applying of the Laws of Logs
Recall that, in chemistry, [latex]\text{pH}=-\mathrm{log}\left[{H}^{+}\right][/latex]. If the concentration of hydrogen ions in a liquid is doubled, what is the effect on pH?
Solution
Suppose C is the original concentration of hydrogen ions, and P is the original pH of the liquid. Then [latex]P=-\mathrm{log}\left(C\right)[/latex]. If the concentration is doubled, the new concentration is 2C. Then the pH of the new liquid is
Using the product rule of logs
Since [latex]P=-\mathrm{log}\left(C\right)[/latex], the new pH is
When the concentration of hydrogen ions is doubled, the pH decreases by about 0.301.
Try It 12
How does the pH change when the concentration of positive hydrogen ions is decreased by half?
SolutionLicenses & Attributions
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- Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..