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Section 7.4 Properties of Logarithms

Subsection 1. Apply the distributive law

We have met several types of algebraic properties before treating logarithms. Here is a review of the most common ones.

Subsubsection Example

Example 7.39.

Which equation is a correct application of the distributive law?

  1. \(2(5 \cdot 3^x) = 10 \cdot 6^x~~~~~\) or \(~~~~~2(5+3^x) = 10+2 \cdot 3^x\)
  2. \(\log (x+10) = \log {x} + \log {10}~~~~~\) or \(~~~~~\dfrac{1}{x}(x+10) = 1 +\dfrac{10}{x}\)
Solution

  1. The distributive law applies to multiplying a sum or difference, not a product. In the first equation, \(5 \cdot 3^x\) is a product, so the distributive law does not apply. (We can, however, simplfy that expression with the associative law:

    \begin{equation*} 2(5 \cdot 3^x) = (2\cdot 5)\cdot 3^x = 10 \cdot 3^x \end{equation*}

    The second equation is a correct application of the distributive law. You can check that the first equation is false and the second equation is true by substituting \(x=1\text{.}\)

  2. The distributive law applies only to multiplying a sum or product, not to other operations, such as taking logs. You can check that the first equation is false by substituting \(x=10\text{.}\)

    The second equation is a correct application of the distributive law.

Subsubsection Exercises

Decide whether each equation is a correct application of the distributive law. Write a correct statement if possible.

\(\dfrac{x+6}{3} \rightarrow \dfrac{x}{3} + \dfrac{6}{3}\)
Answer

Correct

\(\dfrac{6}{x+3} \rightarrow \dfrac{6}{x} + \dfrac{6}{3}\)
Answer

Not correct

\(2(P_0 a^t) \rightarrow 2P_0 +2a^t\)
Answer
Not correct. \(~~2(P_0 a^t) = 2P_0 a^t\)
\(25(1+r)^8 \rightarrow (25+25r)^8\)
Answer

Not correct

Subsection 2. Apply the laws of exponents

Be careful to avoid tempting but false operations with exponents.

Subsubsection Example

Example 7.44.

Which equation is a correct application of the laws of exponents?

  1. \(20(1+r)^4 = 20+20r^4~~~~~\) or \(~~~~~(ab^t)^3 = a^3b^{3t}\)
  2. \(2^{t/5} = (2^{1/5})^t~~~~~\) or \(~~~~~6.8(10)^t = 68^t\)
Solution

  1. The first statement is not correct. There is no law that says \((a+b)^n\) is equivalent to \(a^n+b^n\text{,}\) so \((1+r)^4\) is not equivalent to \(1^4+r^4\) or \(1+r^4\text{.}\)

    However, it is true that \((ab)^n = a^nb^n\text{,}\) so in particular the second statement is true:

    \begin{equation*} ~~(ab^t)^3 = a^3(b^t)^3 = a^3b^{3t} \end{equation*}

  2. The first statement is correct. If we start with \((2^{1/5})^t\text{,}\) we can apply the third law, \((a^m)^n = a^{mn}\text{,}\) to find

    \begin{equation*} (2^{1/5})^t = 2^{(1/5)t} = 2^{t/5}\text{.} \end{equation*}

    In the second statement, 6.8 is not raised to power \(t\text{,}\) so we cannot multiply 6.8 times 10.

Subsubsection Exercises

Decide whether each equation is a correct application of the laws of exponents. Write a correct statement if possible.

\(P(1-r)^6 \rightarrow P-Pr^6\)
Answer

Not correct

\(25(2^t) \cdot 4(2^t) \rightarrow 100 \cdot 2^{t^2}\)
Answer
Not correct. \(~25(2^t) \cdot 4(2^t) = 100(2^{2t})\)
\(a\left(b^{1/8}\right)^{2t} \rightarrow ab^{t/4}\)
Answer

Correct

\(N(0.94)^{1/8.3} \rightarrow \dfrac{N}{(0.94)^{8.3}}\)
Answer
Not correct, but \(~N(0.94)^{-8.3} = \dfrac{N}{(0.94)^{8.3}}\)

Subsection 3. Apply the properties of radicals

Rules for Radicals.

Product Rule

\begin{equation*} \sqrt[n]{ab}=\sqrt[n]{a}~\sqrt[n]{b}~~~~~~~\text{for } a, b \ge 0 \end{equation*}

Quotient Rule

\begin{equation*} \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}~~~~~~~\text{for } a\ge 0,~~ b \gt 0 \end{equation*}

In general, it is not true that \(\sqrt[n]{a+b}\) is equivalent to \(\sqrt[n]{a}+\sqrt[n]{b}\text{,}\) or that \(\sqrt[n]{a-b}\) is equivalent to \(\sqrt[n]{a}-\sqrt[n]{b}\text{.}\)

Subsubsection Examples

Example 7.49.

Which equation is a correct application of the properties of radicals?

  1. \(\sqrt{x^4+81} = x^2 + 9~~~~~\) or \(~~~~~\sqrt[3]{P^2}~\sqrt[3]{1+r} = \sqrt[3]{P^2(1+r)}\)
  2. \(\dfrac{\sqrt{x+y}}{\sqrt{x}} = \sqrt{y}~~~~~\) or \(~~~~~\dfrac{x+y}{\sqrt{x+y}} = \sqrt{x+y}\)
Solution

  1. The first statement is incorrect. There is no property that says \(~\sqrt[n]{a+b} = \sqrt[n]{a} + \sqrt[n]{b}\text{.}\)

    However, it is true that \(~\sqrt[n]{a}~ \sqrt[n]{b} = \sqrt[n]{ab}~\text{,}\) so the second statement is correct.

  2. The first statement is incorrect, because \(\dfrac{x+y}{x}\) is not equivalent to \(y\text{.}\)

    The second statement is correct, because \(~\sqrt{x+y}~\sqrt{x+y} = x+y\text{.}\)

Subsubsection Exercises

Decide whether each equation is a correct application of the properties of radicals. Write a correct statement if possible.

\(\sqrt[4]{a^2-a^4} \rightarrow \sqrt[4]{a^2} - a\)
Answer

Not correct

\(\sqrt{b^4-16} \rightarrow \sqrt{b^2-4} \sqrt{b^2+4}\)
Answer

Correct

\(\sqrt[3]{t^4} + \sqrt[3]{t^4} \rightarrow \sqrt[3]{2t^4}\)
Answer
Not correct.\(~\sqrt[3]{t^4} + \sqrt[3]{t^4} = 2\sqrt[3]{t^4}\)
\(\dfrac{\sqrt{2p}}{\sqrt{4p+8p^2}} \rightarrow \dfrac{1}{\sqrt{2+4p}}\)
Answer

Correct