# Condensing Logarithms – Properties, Explanation, and Examples

Condensing logarithms are helpful when we’re given a long logarithmic expression haring similar bases. This helps us simplify expressions size-wise and save space by combining the expressions that share common bases.

Condensing logarithmic expressions is the process of using different logarithmic properties to combine different logarithmic terms into one quantity.

This article makes use of various concepts we’ve learned in the past, so make sure to review these topics on logarithms before diving right into our main topic – condensing logarithms.

The next section will show you how condensing logarithms is the opposite of expanding logarithms.

How to condense logarithms?

When condensing logarithms, our goal is to compress the expressions altogether by using different logarithmic properties.

\begin{aligned}\color{blue} \log_4 6 + 3\log_4 x – 3\log_4 y &\Leftrightarrow \color{green} \log_2 \left(\dfrac{6x^3}{y^3} \right ) \\ \phantom{xxxxx} \color{blue}\text{Expanded} &\Leftrightarrow \color{green} \boldsymbol{\text{Condensed}}\end{aligned}

This is also why condensing logarithms is the reverse of expanding logarithmic expressions. The example above shows how the two processes are the opposite of each other.

Here are some of the helpful rules that we might need to compress or condense logarithmic expressions.

 Rule Name Algebraic Expression Product Rule $\log_b({\color{blue} A}{\color{green} B}) = \log_b {\color{blue} A} + \log_b {\color{green} B}$ Quotient Rule $\log_b\left(\dfrac{\color{blue} A}{\color{green} B}\right) = \log_b {\color{blue} A} – \log_b {\color{green} B}$ Product Rule $\log_b {\color{blue}A}^n = n\log_b {\color{blue}A}$ Identity Rule $\log_b b = 1$ Zero Rule $\log_b 1 = 0$

To avoid getting overwhelmed with the different logarithmic properties, here are some helpful pointers to look out for:

• Whenever a factor is found outside the logarithm, see if you can apply the power rule right away.
• When combining the terms in between a subtraction or addition operation, check-in the quotient or product rule applies.
• Simplify $\log_1$ or $\log_b b$ using the zero or identity rules.
• The final answer is normally in terms of one rational expression, so double-check when you’re left with extra logarithmic terms.

The examples below will show you the common types of problems that involve condensing logarithms.

Example 1

Condense the logarithmic expression $\log_3 x + \log_3y – \log_3 z$ into a single logarithm.

Solution

Let’s group the terms that are to be added up first, then condense them by using the product rule of logarithms.

\begin{aligned}\log_3 x + \log_3y – \log_3 z&= (\log_3 x + \log_3y )- \log_3 z\\&= \log_3 xy – \log_3 z \color{green} \text{ Product Rule}\end{aligned}

To further condense the expression into one single logarithm, let’s apply the quotient rule for logarithms.

\begin{aligned}\log_3 xy – \log_3 z &= \log_3 \dfrac{xy}{z} \color{green} \text{ Quotient Rule}\end{aligned}

This means that $\log_3 x + \log_3y – \log_3 z$ can be condensed into $\log_3 \dfrac{xy}{z}$.

Example 2

Condense the logarithmic expression $2\ln x – \dfrac{1}{2} \ln y$ into a single logarithm.

Solution