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Section 4.3 Logarithms

Subsection 1. Convert between radicals and powers

Subsubsection Examples

Because a logarithm is an exponent, it is helpful to convert easily between radical notation and exponent noatation.

Example 4.25.

Write each power as a radical.

  1. \(\displaystyle x^{2/3}\)
  2. \(\displaystyle t^{-3/2}\)
  3. \(\displaystyle w^{1.25}\)
  4. \(\displaystyle z^{-0.4}\)
Solution

Recall that the numerator of the exponent is the power and the denominator is the root. A negative exponent indicates a reciprocal.

  1. \(\displaystyle x^{2/3} = \sqrt[3]{x^2}\)
  2. \(\displaystyle t^{-3/2} = \dfrac{1}{t^{3/2}} = \dfrac{1}{\sqrt{t^3}}\)
  3. \(\displaystyle w^{1.25} = w^{5/4} = \sqrt[4]{w^5}\)
  4. \(\displaystyle z^{-0.4} = \dfrac{1}{z^{4/10}} = \dfrac{1}{z^{2/5}} = \dfrac{1}{\sqrt[5]{z^2}}\)
Example 4.26.

Write each radical expression in exponential form and simplify.

  1. \(\displaystyle \sqrt[4]{b^3}\)
  2. \(\displaystyle \dfrac{1}{\sqrt[6]{a^3}}\)
  3. \(\displaystyle x^2\sqrt[4]{x}\)
  4. \(\displaystyle \dfrac{\sqrt[3]{v}}{\sqrt{v}}\)
Solution
  1. \(\displaystyle \sqrt[4]{b^3} = b^{3/4}\)
  2. \(\displaystyle \dfrac{1}{\sqrt[6]{a^3}} = \dfrac{1}{a^{3/6}} = a^{-3/6} = a^{-1/2}\)
  3. \(\displaystyle x^2\sqrt[4]{x} = x^2x^{1/4} = a^{2+1/4} = a^{9/4}\)
  4. \(\displaystyle \dfrac{\sqrt[3]{v}}{\sqrt{v}} = \dfrac{v^{1/3}}{v^{1/2}} = v^{1/3-1/2} = v^{-1/6}\)

Subsubsection Exercises

Write each power as a radical.

  1. \(\displaystyle m^{-3/5}\)
  2. \(\displaystyle p^{2.75}\)
  3. \(\displaystyle x^{0.18}\)
Answer
  1. \(\displaystyle \dfrac{1}{\sqrt[5]{m^3}}\)
  2. \(\displaystyle \sqrt[4]{p^{11}}\)
  3. \(\displaystyle \sqrt[50]{x^{9}}\)

Write each radical expression in exponential form and simplify.

  1. \(\displaystyle \sqrt[10]{n^9}\)
  2. \(\displaystyle \sqrt{h}\sqrt[4]{h}\)
  3. \(\displaystyle \left(\sqrt[3]{t^2}\right)^4\)
Answer
  1. \(\displaystyle n^{9/10}\)
  2. \(\displaystyle h^{3/4}\)
  3. \(\displaystyle t^{8/3}\)

Subsection 2. Find an unknown exponent

If we can write both sides of an equation as powers with the same base, we can equate the exponents.

Subsubsection Examples

Example 4.29.

Find the value of the exponent.

  1. \(\displaystyle 3^x=81\)
  2. \(\displaystyle 5^x=\dfrac{1}{125}\)
  3. \(\displaystyle \left(\dfrac{3}{4}\right)^x=\dfrac{16}{9}\)
  4. \(\displaystyle 64^x=16\)
Solution
  1. We can write both sides with base 3.

    \begin{equation*} ~~ 81 = 3^4,~~~\text{so} ~~x=4\text{.} \end{equation*}
  2. We can write both sides with base 5.

    \begin{equation*} ~~125 = 5^3,~~~ \text{so}~~~5^{-3} = \dfrac{1}{125},~~~ \text{and} ~~x=-3\text{.} \end{equation*}
  3. \(\left(\dfrac{3}{4}\right)^2 = \dfrac{9}{16}~\text{,}\) so \(~\left(\dfrac{3}{4}\right)^{-2} = \dfrac{16}{9}~~\text{,}\) and \(~x=-2\text{.}\)
  4. We can write both sides with base 4.

    \begin{equation*} 64 = 4^3~~~ \text{and}~~~16 = 4^2\text{,} \end{equation*}

    so

    \begin{align*} ~(4^3)^x \amp = 4^2 \amp\amp \blert{\text{Multiply exponents.}}\\ 4^{3x} \amp = 4^2 \amp\amp \blert{\text{Equate exponents.}}\\ 3x \amp = 2\\ x \amp =\dfrac{2}{3} \end{align*}
Example 4.30.

By using trial and error, estimate the value of the exponent to the nearest tenth.

  1. \(\displaystyle 2^x=15\)
  2. \(\displaystyle 3^x=65\)
  3. \(\displaystyle 10^x=0.03\)
  4. \(\displaystyle 0.5^x = 0.20\)
Solution
  1. \(2^4=16~\text{,}\) so we try a slightly smaller exponent and find that \(2^{3.9} = 14.9285~\text{,}\) so \(x \approx 3.9\text{.}\)
  2. 65 is between \(3^3=27\) and \(3^4=81\text{,}\) so \(x\) must be between 3 and 4. By trying exponents 3.1, 3.2, 3.3, and so on, we find that \(3^{3.8}=65.022\text{,}\) so \(x \approx 3.8\text{.}\)
  3. \(10^{-1}=0.1\) and \(10^{-2}=0.01\text{,}\) so \(-2 \lt x \lt -1\text{.}\) By trying exponents between \(-2\) and \(-1\text{,}\) we find that \(10^{-1.5} = 0.0316\text{,}\) so \(x \approx -1.5\text{.}\)
  4. \(0.5^2 = 0.25\text{,}\) and as we increase the exponent on \(0.5\text{,}\) the result will be smaller. By trial and error we find that \(0.5^{2.3} = 0.2031\text{,}\) so \(x \approx 2.3\text{.}\)

Subsubsection Exercises

Find the value of the exponent.

  1. \(\displaystyle 2^x = \dfrac{1}{1024}\)
  2. \(\displaystyle 125^x = 25\)
Answer
  1. \(\displaystyle 10\)
  2. \(\displaystyle \dfrac{2}{3}\)

By using trial and error, estimate the value of the exponent to the nearest tenth.

  1. \(\displaystyle 10^x = 50\)
  2. \(\displaystyle 1.08^x = 1.5\)
Answer
  1. \(\displaystyle 1.7\)
  2. \(\displaystyle 5.3\)

Subsection 3. Apply the laws of exponents

The laws of exponents still apply to variable exponents. (If you would like to review the laws of exponents, they are listed in Section 3.2.)

Subsubsection Examples

Example 4.33.

Use the laws of exponents to simplify.

  1. \(\displaystyle 1.35^6(1.35^4)\)
  2. \(\displaystyle 0.64^5(0.64^n)\)
Solution

When multiplying two powers with the same base, we add the exponents. Notice that the base does not change.

  1. \(\displaystyle 1.35^6(1.35^4) = 1.35^{6+4}=1.35^{10}\)
  2. \(\displaystyle 0.64^5(0.64^n) = 0.64^{5+n}\)
Example 4.34.

Use the laws of exponents to simplify.

  1. \(\displaystyle \dfrac{0.32^8}{0.32^2}\)
  2. \(\displaystyle \dfrac{0.32^t}{0.32^x}\)
Solution

When dividing two powers with the same base, we subtract the exponents.

  1. \(\displaystyle \dfrac{0.32^8}{0.32^2} = 0.32^{8-2} = 0.32^6\)
  2. \(\displaystyle \dfrac{0.32^t}{0.32^x} = 0.32^{t-x}\)
Example 4.35.

Use the laws of exponents to simplify.

  1. \(\displaystyle (1.07^5)^3\)
  2. \(\displaystyle (1.07^4)^p\)
Solution

When raising a power to a power, we multiply the exponents.

  1. \(\displaystyle (1.07^5)^3 = 1.07^{15}\)
  2. \(\displaystyle (1.07^4)^p = 1.07^{4p}\)

Subsubsection Exercise

Use the laws of exponents to simplify \(~2.5^{2t}(2.5^3)\text{.}\)

Answer
\(2.5^{2t+3}\)

Use the laws of exponents to simplify \(~(0.94^4)^{m-2}\text{.}\)

Answer
\(0.94^{4m-8}\)

Use the laws of exponents to simplify \(~\dfrac{1.13^{8x}}{1.13^{5x}}\text{.}\)

Answer
\(1.13^{3x}\)