4.3 Laws of Logarithms
4.3.1 Learning Objectives
By the end of this section, you should be able to:
- apply the product, quotient, and power laws;
- expand logarithmic expressions;
- combine logarithms into a single expression;
- recognize invalid uses of logarithm laws; and
- evaluate logarithms using the change-of-base formula.
4.3.2 Product, Quotient, and Power Laws
For positive \(M\) and \(N\),
\[ \log_a(MN)=\log_aM+\log_aN, \]
\[ \log_a\left(\frac MN\right)=\log_aM-\log_aN, \]
and, for real \(p\),
\[ \log_a(M^p)=p\log_aM. \]
There is no corresponding rule for the logarithm of a sum or difference.
Example 4.11 Applying the Logarithm Laws
Evaluate \(\log_2 8+\log_2 4-\log_2 2\).
Solution
\[ \log_2\left(\frac{8\cdot4}{2}\right)=\log_2 16=4. \]
\[ \boxed{4} \]
Exercise 4.11 Evaluate \(\log_3 27-\log_3 3\).
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Answer: \(\log_3(27/3)=\log_3 9=\boxed{2}\).
4.3.3 Expanding Logarithmic Expressions
Expanding rewrites one logarithm as a sum or difference. Factor expressions when useful, apply the product or quotient law, then use the power law.
Example 4.12 Expanding a Logarithm
Expand
\[ \ln\left(\frac{x^3\sqrt{x+1}}{(x-2)^2}\right), \]
assuming all quantities are positive.
Solution
\[ \boxed{3\ln x+\frac12\ln(x+1)-2\ln(x-2)} \]
Exercise 4.12 Expand \(\log_5\left(\dfrac{a^2b}{c^4}\right)\), assuming \(a,b,c>0\).
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Answer: \(\boxed{2\log_5a+\log_5b-4\log_5c}\)
4.3.4 Combining Logarithmic Expressions
Combining reverses expansion. Move coefficients into exponents, combine sums as products, and combine differences as quotients.
Example 4.13 Writing a Single Logarithm
Combine
\[ 2\ln x-\frac12\ln y+\ln3. \]
Solution
\[ 2\ln x=\ln(x^2),\qquad \frac12\ln y=\ln(\sqrt y). \]
Therefore,
\[ \boxed{\ln\left(\frac{3x^2}{\sqrt y}\right)}. \]
Exercise 4.13 Combine \(\log a+\log b-3\log c\).
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Answer: \(\boxed{\log\left(\dfrac{ab}{c^3}\right)}\)
4.3.5 Change of Base
Calculators directly evaluate common and natural logarithms. Any other base can be evaluated using
\[ \log_a x=\frac{\log x}{\log a}=\frac{\ln x}{\ln a}. \]
Example 4.14 Evaluating with Change of Base
Evaluate \(\log_7 20\) to four decimal places.
Solution
\[ \log_7 20=\frac{\ln20}{\ln7}\approx1.5395. \]
\[ \boxed{1.5395} \]
Exercise 4.14 Evaluate \(\log_3 14\) to three decimal places.
Show answer
Answer: \(\boxed{\log_3 14=\dfrac{\ln14}{\ln3}\approx2.402}\)
4.3.6 Conceptual Takeaways
- Logarithm laws come from exponent laws.
- Products become sums, quotients become differences, and powers become coefficients.
- Logarithms do not distribute across addition or subtraction.
- Expanding and combining are inverse algebraic processes.
- Change of base allows any logarithm to be evaluated on a calculator.
4.3.7 Skills You Should Be Able to Do
- Apply all three logarithm laws.
- Expand logarithmic expressions.
- Combine several logarithms.
- Identify invalid logarithmic manipulations.
- Use change of base accurately.
4.3.8 Practice Problems with Solutions
Expand \(\log_2(8x)\).
Show Solution
\[ \log_2(8x)=\log_28+\log_2x=\boxed{3+\log_2x}. \]
Expand \(\ln(x^4/y^3)\), assuming \(x,y>0\).
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\[ \ln(x^4/y^3)=\ln(x^4)-\ln(y^3)=\boxed{4\ln x-3\ln y}. \]
Combine \(2\log x+\log y\).
Show Solution
\[ 2\log x=\log(x^2), \]
so \(\boxed{\log(x^2y)}\).
Combine \(\ln a-\ln b+\frac12\ln c\).
Show Solution
\[ \boxed{\ln\left(\frac{a\sqrt c}{b}\right)}. \]
Explain why \(\log(x+4)\ne\log x+\log4\).
Show Solution
The product law applies to multiplication, not addition. For example, at \(x=6\), the left side is \(\log10=1\), while the right side is \(\log24\ne1\). Therefore, \(\boxed{\text{the proposed identity is false}}\).
Evaluate \(\log_6 50\) to three decimal places.
Show Solution
\[ \log_650=\frac{\ln50}{\ln6}\approx2.183. \]
\[ \boxed{2.183} \]