Paul's Online Notes
Home / Algebra Trig Review / Exponentials & Logarithms / Simplifying Logarithms
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 3.4 : Simplifying Logarithms

Simplify each of the following logarithms. Show All Solutions Hide All Solutions

1. $$\ln {x^3}{y^4}{z^5}$$
Show Solution

Here simplify means use Property 1 - 7 from the Logarithm Properties section as often as you can. You will be done when you can’t use any more of these properties.

Property 5 can be extended to products of more than two functions so,

\begin{align*}\ln {x^3}{y^4}{z^5} & = \ln {x^3} + \ln {y^4} + \ln {z^5}\\ & = 3\ln x + 4\ln y + 5\ln z\end{align*}
2. $${\log _3}\left( {\frac{{9{x^4}}}{{\sqrt y }}} \right)$$
Show Solution

In using property 6 make sure that the logarithm that you subtract is the one that contains the denominator as its argument. Also, note that I’ll be converting the root to exponents in the first step since we’ll need that done for a later step.

\begin{align*}{\log _3}\left( {\frac{{9{x^4}}}{{\sqrt y }}} \right) & = {\log _3}9{x^4} - {\log _3}{y^{\frac{1}{2}}}\\ & = {\log _3}9 + {\log _3}{x^4} - {\log _3}{y^{\frac{1}{2}}}\\ & = 2 + 4{\log _3}x - \frac{1}{2}{\log _3}y\end{align*}

Evaluate logs where possible as I did in the first term.

3. $$\log \left( {\frac{{{x^2} + {y^2}}}{{{{\left( {x - y} \right)}^3}}}} \right)$$
Show Solution

The point to this problem is mostly the correct use of property 7.

\begin{align*}\log \left( {\frac{{{x^2} + {y^2}}}{{{{\left( {x - y} \right)}^3}}}} \right) & = \log \left( {{x^2} + {y^2}} \right) - \log {\left( {x - y} \right)^3}\\ & = \log \left( {{x^2} + {y^2}} \right) - 3\log \left( {x - y} \right)\end{align*}

You can use Property 7 on the second term because the WHOLE term was raised to the 3, but in the first logarithm, only the individual terms were squared and not the term as a whole so the 2’s must stay where they are!