ELF - Applying Properties of Logarithms Lesson

Applying Properties of Logarithms

In the previous lesson, we explored the concept of "estimating" or "approximating" the common log of a value or the natural log of a value, but what happens when the base of the log is not "10" or not "e?" What happens in the situation where we have two logarithmic expressions being added, subtracted, multiplied, or divided? What we do in these situations?

In this lesson we will explore various properties of logarithms, as well as the change-of-base formula (see below), which will allow us to simplify and evaluate various logarithms that do not always have a base of "10" or "e."

Rule	Formula
Product Rule	$LaTeX: \log_b\left(xy\right)=\log_bx+\log_by$ $\log_b\left(xy\right)=\log_bx+\log_by$
Quotient Rule	$LaTeX: \log_b\left(\frac{x}{y}\right)=\log_bx-\log_by$ $\log_b\left(\frac{x}{y}\right)=\log_bx-\log_by$
Power Rule	$LaTeX: \log_bx^r=r\log b^2$ $\log_bx^r=r\log b^2$
Change of Base Rule	$LaTeX: \log_ax=\frac{\log_bx}{\log_ba}$ $\log_ax=\frac{\log_bx}{\log_ba}$
Equality Rule	$LaTeX: If\:\log_ax=\log_ay,\:then\:x=y$ $If\:\log_ax=\log_ay,\:then\:x=y$

Example of Natural Logarithmic Properties

$LaTeX: \log_ee=\ln\:e=e \\ \log_e1=\ln\:1=0 \\ e^{\log_{e^6}}=e^{\ln\:6}=6 \\ \log_ee^3=\ln\:e^3=3$ $\log_ee=\ln\:e=e \\ \log_e1=\ln\:1=0 \\ e^{\log_{e^6}}=e^{\ln\:6}=6 \\ \log_ee^3=\ln\:e^3=3$

Properties of Logarithms

We will first explore properties of logarithms, condensing and expanding logarithms, which will allow us to apply of these properties of logarithms. The following visual examples show the properties of logarithms "in action."

Example 1

$LaTeX: 2\left(\log6+\log x\right)-\log12\left[Product\:Property\right] \\ 2\left(\log6x\right)\log12\left[Power\:Property\right] \\ \log6x^2-\log12 \\ \log36x^2-\log12\:\left[Quotient\:Property\right] \\ \log\frac{36x^2}{12} \\ \log3x^2$ $2\left(\log6+\log x\right)-\log12\left[Product\:Property\right] \\ 2\left(\log6x\right)\log12\left[Power\:Property\right] \\ \log6x^2-\log12 \\ \log36x^2-\log12\:\left[Quotient\:Property\right] \\ \log\frac{36x^2}{12} \\ \log3x^2$

Example 2

$LaTeX: \log_2\frac{7x^3}{y}= \\ \log_27x^3-\log_2y= \\ \log_27+\log_2x^3-\log_2y= \\ \log_27+3\log_2x-\log_2y$ $\log_2\frac{7x^3}{y}= \\ \log_27x^3-\log_2y= \\ \log_27+\log_2x^3-\log_2y= \\ \log_27+3\log_2x-\log_2y$

Condensing Example	Solution
$LaTeX: \frac{1}{4}\cdot\log_2y-7\cdot\log_2z$ $\frac{1}{4}\cdot\log_2y-7\cdot\log_2z$	$LaTeX: \log_2\frac{\sqrt[4]{y}}{z^7}$ $\log_2\frac{\sqrt[4]{y}}{z^7}$
$LaTeX: \log_5x^3-\log_5y+\frac{1}{4}\cdot\log_514-3\cdot\log_5t$ $\log_5x^3-\log_5y+\frac{1}{4}\cdot\log_514-3\cdot\log_5t$	$LaTeX: \log_5\left(\frac{x^3\cdot\sqrt[4]{14}}{y\cdot t^3}\right)$ $\log_5\left(\frac{x^3\cdot\sqrt[4]{14}}{y\cdot t^3}\right)$
$LaTeX: 4\log_{11}x-\left(5\cdot\log_{11}t+4\cdot\log_{11}y\right)$ $4\log_{11}x-\left(5\cdot\log_{11}t+4\cdot\log_{11}y\right)$	$LaTeX: \log_{11}\left(\frac{x^4}{t^5y^4}\right)$ $\log_{11}\left(\frac{x^4}{t^5y^4}\right)$
$LaTeX: \left(\log_2u+\log_2v-5\cdot\log_2y\right)$ $\left(\log_2u+\log_2v-5\cdot\log_2y\right)$	$LaTeX: \log_2\left(\frac{uv}{y^5}\right)$ $\log_2\left(\frac{uv}{y^5}\right)$
$LaTeX: \log_3\sqrt[]{27}-\log_5\sqrt[]{5}$ $\log_3\sqrt[]{27}-\log_5\sqrt[]{5}$	$LaTeX: \log_327^{\frac{1}{2}}-\log_55^{\frac{1}{2}} \\ \frac{1}{2}\log_327-\frac{1}{2}\log_55 \\ \frac{1}{2}\left(3\right)-\frac{1}{2}\left(1\right) \\ \frac{3}{2}-\frac{1}{2}=\frac{2}{2}=1$ $\log_327^{\frac{1}{2}}-\log_55^{\frac{1}{2}} \\ \frac{1}{2}\log_327-\frac{1}{2}\log_55 \\ \frac{1}{2}\left(3\right)-\frac{1}{2}\left(1\right) \\ \frac{3}{2}-\frac{1}{2}=\frac{2}{2}=1$
$LaTeX: \ln6e^5-\ln7$ $\ln6e^5-\ln7$	$LaTeX: \ln6+\ln e^5-\ln7 \\ \ln6+\left(5\right)\ln e-\ln7 \\ 1.7918+5+1.9459=8.7377$ $\ln6+\ln e^5-\ln7 \\ \ln6+\left(5\right)\ln e-\ln7 \\ 1.7918+5+1.9459=8.7377$

Next, watch the following videos to explore and apply the properties of logarithms. Be very meticulous in taking notes as you watch these videos.

Change of Base Formula

We have already explored the change-of-base formula, using common logarithms, but now that we know more about natural logarithms we can also use the change-of-base formula with natural logarithms.

Now, we will watch the videos below which will help us in exploring the change-of-base formula. We will also explore the applications of logarithms using the change-of-base formula. Be very meticulous in taking notes as you watch these videos.

The following give visual examples of the Change-of-Base Formula.

$LaTeX: Change\:of\:Base\:Formula \\ \log_ba=\frac{\log_ca}{\log_cb} \\ \ast \log_ba=x\:so\:b^x=a,\:therefore\:b^x=a \\ \ast \log_ca=y\:so\:c^y=a,\:therefore\:\left(c^z\right)^x=c^y \\ \ast \log_cb=z\:so\:c^z=b,\:therefore\:c^{zx}=y\:and\:or\:zx=y\:and\:or\:x=\frac{y}{z}$ $Change\:of\:Base\:Formula \\ \log_ba=\frac{\log_ca}{\log_cb} \\ \ast \log_ba=x\:so\:b^x=a,\:therefore\:b^x=a \\ \ast \log_ca=y\:so\:c^y=a,\:therefore\:\left(c^z\right)^x=c^y \\ \ast \log_cb=z\:so\:c^z=b,\:therefore\:c^{zx}=y\:and\:or\:zx=y\:and\:or\:x=\frac{y}{z}$

$LaTeX: \log_425=\frac{\log25}{\log4}=\frac{1.39794}{.060206}=2.3219$ $\log_425=\frac{\log25}{\log4}=\frac{1.39794}{.060206}=2.3219$

$LaTeX: \log_212=\frac{\log12}{\log2}=\frac{1.07918}{.30103}=3.5850$ $\log_212=\frac{\log12}{\log2}=\frac{1.07918}{.30103}=3.5850$

Now that we have explored these properties and formula, it's now time for us to explore and practice working with the properties of logarithms, as well as the change-of-base formula.

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