Skip to main content
Mathematics LibreTexts

3.4: Algebra Tips and Tricks Part VI (Logarithms)

  • Page ID
    88644
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Logarithms

    A logarithm is the inverse function to an exponential function. For example, for the exponential function \(y = 2^x\), if we have an input of \(x = 6\), we get an output of \(y = 64\), and we write \(64 = 2^6\). The logarithmic function \(y = \log_2(x)\) is the reverse of this. We swap the input and the output, so now \(x = 64\) and \(y = 6\). We see \(6 = \log_2(64)\).

    In calculus, we will mostly use the exponential function \(e^x\) and its inverse, \(\ln(x)\). Below are some important formulas:

    \[\begin{align*} e^{\ln(x)} & = x \\ \ln(e^x) & = x \\ \ln(x) + \ln(y) & = \ln(xy) \\ \ln(x) - \ln(y) & = \ln\left(\frac{x}{y}\right) \\ a \ln(x) & = \ln(x^a) \end{align*}\]

    Examples:

    \(\ln(x^2) - \ln(x)\).

    There are two ways to do this one. First, we can bring down the exponent of two down in front \(\ln(x^2) = 2 \ln(x)\). Then can combine the like terms of \(2\ln(x)\) and \(\ln(x)\):

    \[\begin{align*} \ln(x^2) - \ln(x) & = 2 \ln(x) - \ln(x) \\ & = \boxed{\ln(x)} \end{align*}\]

    Alternatively, we can rewrite the subtraction as a division, like so:

    \[\begin{align*} \ln(x^2) - \ln(x) & = \ln\left(\frac{x^2}{x}\right) \\ & = \boxed{\ln(x)} \end{align*}\]

    Either way we get the same answer!

    \(\ln(e^3 x^4) - 3 \ln(x)\).

    First, we rewrite the multiplication using addition. Then we can simply from there.

    \[\begin{align*} \ln(e^3 x^4) - 3 \ln(x) & = \ln(e^3) + \ln(x^4) - 3 \ln(x)\\ & = 3 + 4 \ln(x) - 3 \ln(x) \\ & = \boxed{3 + \ln(x)} \end{align*}\]

    \(\ln(\sqrt{x})\).

    We know quicklatex.com-03ca0b7d849a68cd8b1f113474f07f06_l3.png, so .

    \(\ln\left(\frac{\sqrt{x} y}{z^3}\right) - \ln\left(\frac{z}{\sqrt{x} y^3}\right)\).

    We can rewrite all the products and divisions as addition and subtraction:

    \[\begin{align*} \ln\left(\frac{\sqrt{x} y}{z^3}\right) - \ln\left(\frac{z}{\sqrt{x} y^3}\right) & = \ln(\sqrt{x}) + \ln(y) - \ln(z^3) - [\ln(z) - \ln(\sqrt{x}) - \ln(y^3)] \\ & = \frac{1}{2} \ln(x) + \ln(y) - 3 \ln(z) - \ln(z) + \frac{1}{2} \ln(x) + 3 \ln(y) \\ & = \boxed{\ln(x) + 4 \ln(y) - 4 \ln(z)}. \end{align*}\]


    This page titled 3.4: Algebra Tips and Tricks Part VI (Logarithms) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Tyler Seacrest via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.