In Exercises expandlogfirst - expandloglast, expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers.
\(\ln (x^{3}y^{2})\)

\(3\ln (x) + 2\ln (y)\)
\(\log _{2}\left (\frac {128}{x^{2} + 4}\right )\)

\(7 - \log _{2}(x^{2} + 4)\)
\(\log _{5}\left (\frac {z}{25}\right )^{3}\)

\(3\log _{5}(z) - 6\)
\(\log (1.23 \times 10^{37})\)

\(\log (1.23) + 37\)
\(\ln \left (\frac {\sqrt {z}}{xy}\right )\)

\(\frac {1}{2}\ln (z) - \ln (x) - \ln (y)\)
\(\log _{5} \left (x^2 - 25 \right )\)

\(\log _{5}(x-5) + \log _{5}(x+5)\)
\(\log _{\sqrt {2}} \left (4x^3\right )\)

\(3\log _{\sqrt {2}}(x) + 4\)
\(\log _{\frac {1}{3}}(9x(y^{3} - 8))\)

\(-2 + \log _{\frac {1}{3}}(x) + \log _{\frac {1}{3}}(y - 2) + \log _{\frac {1}{3}}(y^{2} + 2y + 4)\)
\(\log \left (1000x^3y^5\right )\)

\(3 + 3\log (x) + 5 \log (y)\)
\(\log _{3} \left (\frac {x^2}{81y^4}\right )\)

\(2\log _{3}(x) - 4 - 4\log _{3}(y)\)
\(\ln \left (\sqrt [4]{\frac {xy}{ez}}\right )\)

\(\frac {1}{4} \ln (x) + \frac {1}{4} \ln (y) - \frac {1}{4} - \frac {1}{4} \ln (z)\)
\(\log _{6} \left (\frac {216}{x^3y}\right )^4\)

\(12-12\log _{6}(x) - 4\log _{6}(y)\)
\(\log \left (\frac {100x\sqrt {y}}{\sqrt [3]{10}}\right )\)

\(\frac {5}{3}+\log (x)+\frac {1}{2}\log (y)\)
\(\log _{\frac {1}{2}}\left (\frac {4\sqrt [3]{x^2}}{y\sqrt {z}}\right )\)

\(-2+\frac {2}{3}\log _{\frac {1}{2}}(x)-\log _{\frac {1}{2}}(y)-\frac {1}{2}\log _{\frac {1}{2}}(z)\)
\(\ln \left (\frac {\sqrt [3]{x}}{10 \sqrt {yz}}\right )\)

\(\frac {1}{3} \ln (x) - \ln (10) - \frac {1}{2}\ln (y)-\frac {1}{2}\ln (z)\)

In Exercises combinelogfirst - combineloglast, use the properties of logarithms to write the expression as a single logarithm.

\(4\ln (x) + 2\ln (y)\)

\(\ln (x^{4}y^{2})\)
\(\log _{2}(x) + \log _{2}(y) - \log _{2}(z)\)

\(\log _{2}\left (\frac {xy}{z}\right )\)
\(\log _{3}(x) - 2 \log _{3}(y)\)

\(\log _{3} \left ( \frac {x}{y^2} \right )\)
\(\frac {1}{2}\log _{3}(x) - 2\log _{3}(y) - \log _{3}(z)\)

\(\log _{3}\left (\frac {\sqrt {x}}{y^{2}z}\right )\)
\(2 \ln (x) -3 \ln (y) - 4\ln (z)\)

\(\ln \left ( \frac {x^2}{y^3z^4} \right )\)
\(\log (x) - \frac {1}{3} \log (z) + \frac {1}{2} \log (y)\)

\(\log \left (\frac {x \sqrt {y}}{\sqrt [3]{z}} \right )\)
\(-\frac {1}{3} \ln (x) - \frac {1}{3}\ln (y) + \frac {1}{3} \ln (z)\)

\(\ln \left (\sqrt [3]{\frac {z}{xy}} \right )\)
\(\log _{5}(x) - 3\)

\(\log _{5}\left (\frac {x}{125}\right )\)
\(3 - \log (x)\)

\(\log \left (\frac {1000}{x}\right )\)
\(\log _{7}(x) + \log _{7}(x - 3) - 2\)

\(\log _{7}\left (\frac {x(x - 3)}{49}\right )\)
\(\ln (x) + \frac {1}{2}\)

\(\ln \left (x \sqrt {e} \right )\)
\(\log _{2}(x) + \log _{4}(x)\)

\(\log _{2}\left (x^{3/2}\right )\)
\(\log _{2}(x) + \log _{4}(x-1)\)

\(\log _{2}\left (x \sqrt {x-1}\right )\)
\(\log _{2}(x) + \log _{\frac {1}{2}}(x - 1)\)

\(\log _{2}\left (\frac {x}{x - 1}\right )\)

In Exercises changeofbasefirst - changeofbaselast, use the appropriate change of base formula to convert the given expression to an expression with the indicated base.

\(7^{x - 1}\) to base \(e\)

\(7^{x - 1} = e^{(x - 1)\ln (7)}\)
\(\log _{3}(x + 2)\) to base 10

\(\log _{3}(x + 2) = \frac {\log (x + 2)}{\log (3)}\)
\(\left (\frac {2}{3}\right )^{x}\) to base \(e\)

\(\left (\frac {2}{3}\right )^{x} = e^{x\ln (\frac {2}{3})}\)
\(\log (x^{2} + 1)\) to base \(e\)

\(\log (x^{2} + 1) = \frac {\ln (x^{2} + 1)}{\ln (10)}\)

In Exercises changeofbaseapproxfirst - changeofbaseapproxlast, use the appropriate change of base formula to approximate the logarithm.

\(\log _{3}(12)\)

\(\log _{3}(12) \approx 2.26186\)
\(\log _{5}(80)\)

\(\log _{5}(80) \approx 2.72271\)
\(\log _{6}(72)\)

\(\log _{6}(72) \approx 2.38685\)
\(\log _{4}\left (\frac {1}{10}\right )\)

\(\log _{4}\left (\frac {1}{10}\right ) \approx -1.66096\)
\(\log _{\frac {3}{5}}(1000)\)

\(\log _{\frac {3}{5}}(1000) \approx -13.52273\)
\(\log _{\frac {2}{3}}(50)\)

\(\log _{\frac {2}{3}}(50) \approx -9.64824\)

In Example intrologex number findformulaforlogexample in Section LogarithmicFunctions, we obtained the solution \(F(x) = \log _{2}(-x+4)-3\) as one formula for the given graph by making a simplifying assumption that \(b = -1\). This exercises explores if there are any other solutions for different choices of \(b\).

  1. Show \(G(x) =\log _{2}(-2x+8) - 4\) also fits the data for the given graph.
  2. Use properties of logarithms to show \(G(x) = \log _{2}(-2x+8) -4 = \log _{2}(-x+4)-3 = F(x)\).
  3. With help from your classmates, find solutions to Example intrologex number findformulaforlogexample in Section LogarithmicFunctions by assuming \(b = -4\) and \(b = -8\). In each case, use properties of logarithms to show the solutions reduce to \(F(x) = \log _{2}(-x+4)-3\).

  4. Using properties of logarithms and the fact that the range of \(\log _{2}(x)\) is all real numbers, show that any function of the form \(f(x) = a \log _{2}(bx-h) + k\) where \(a \neq 0\) can be rewritten as:

    \[ f(x) = a \left ( \log _{2}(bx-h) + \frac {k}{a}\right ) = a ( \log _{2}(bx -h) + \log _{2}(p)) = a \log _{2}(p(bx-h)) = a \log _{2}(pbx - ph),\]

    where \(\frac {k}{a} = \log _{2}(p)\) for some positive real number \(p\). Relabeling, we get every function of the form \(f(x) = a \log _{2}(bx-h) + k\) with four parameters (\(a\), \(b\), \(h\), and \(k\)) can be rewritten as \(f(x) = a \log _{2}(Bx - H)\), a formula with just three parameters: \(a\), \(B\), and \(H\).

    Show every solution to Example intrologex number findformulaforlogexample in Section LogarithmicFunctions can be written in the form \(f(x) = \log _{2}\left ( -\frac {1}{8} x + \frac {1}{2} \right )\) and that, in particular, \(F(x) = \log _{2}(-x+4) -3 = \log _{2}\left ( -\frac {1}{8} x + \frac {1}{2} \right ) = f(x)\). Hence, there is really just one solution to Example intrologex number findformulaforlogexample in Section LogarithmicFunctions.

The Henderson-Hasselbalch Equation: Suppose \(HA\) represents a weak acid. Then we have a reversible chemical reaction

\[HA \rightleftharpoons H^{+} + A^{-}.\]
The acid disassociation constant, \(K_{a}\), is given by
\[K_{a} = \frac {[H^{+}][A^{-}]}{[HA]} = [H^{+}]\frac {[A^{-}]}{[HA]},\]
where the square brackets denote the concentrations just as they did in Exercise pHexercise in Section LogarithmicFunctions. The symbol p\(K_{a}\) is defined similarly to pH in that p\(K_{a} = -\log (K_{a})\). Using the definition of pH from Exercise pHexercise and the properties of logarithms, derive the Henderson-Hasselbalch Equation:
\[\mbox {pH} = \mbox {p}K_{a} + \log \frac {[A^{-}]}{[HA]}\]
Compare and contrast the graphs of \(y = \ln (x^{2})\) and \(y = 2\ln (x)\).
Prove the Quotient Rule and Power Rule for Logarithms.
Give numerical examples to show that, in general,
  1. \(\log _{b}(x + y) \neq \log _{b}(x) + \log _{b}(y)\)
  2. \(\log _{b}(x - y) \neq \log _{b}(x) - \log _{b}(y)\)
  1. \(\log _{b}\left (\frac {x}{y}\right ) \neq \frac {\log _{b}(x)}{\log _{b}(y)}\)
Research the history of logarithms including the origin of the word ‘logarithm’ itself. Why is the abbreviation of natural log ‘ln’ and not ‘nl’?
There is a scene in the movie ‘Apollo 13’ in which several people at Mission Control use slide rules to verify a computation. Was that scene accurate? Look for other pop culture references to logarithms and slide rules.

    1. Use properties of logarithm functions to show that if \(f(x) = \ln (x)\), then

      \[ f'(x) = \lim _{h \rightarrow 0} \frac {f(x+h) - f(x)}{h} = \lim _{h \rightarrow 0} \frac {\ln \left ( 1 + \frac {h}{x} \right )}{h} \]
    2. Numerically and graphically investigate the limit: \(\ds {\lim _{h \rightarrow 0}}\) \(\frac {\ln \left ( 1 + \frac {h}{x} \right )}{h}\) for various positive numbers \(x\) to convince yourself that \(\ds {\lim _{h \rightarrow 0}}\) \(\frac {\ln \left ( 1 + \frac {h}{x} \right )}{h} = \frac {1}{x}\)
    3. Use parts diffquotln and speciallnlimit to show to show that the derivative of \(\ln (x)\) is …\(\frac {1}{x}\).

    4. Write the equations of the tangent lines to \(y = \ln (x) \) at the following points. Check your answers graphically. Compare your answers the tangent lines at the corresponding points in Exercise derivativeofex in Section ExponentialFunctions.

      1. \((1,0)\)
      2. \((e, 1)\)
      3. \(\left (e^{-1}, -1 \right )\)