The reverse process of widening logarithms is referred to as combining or condensing logarithmic expressions right into a single quantity. Other textbooks describe this as simplifying logarithms. But, lock all typical the same.

You are watching: Write the expression as a single logarithm

The idea is the you are given a bunch of log in expressions as sums and/or differences, and your task is to put them back or compress right into a “nice” one log in expression.

I extremely recommend that you testimonial the rules of logarithms very first before looking in ~ the worked instances belowbecause you’ll use them in reverse.

For instance, if you walk from left to ideal of the equation climate youmust it is in expanding, while walking from appropriate to left climate you must be condensing.


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Study the description of each ruleto gain an intuitive expertise of it.

Description of every Logarithm Rule


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The logarithm the the product of numbers is the amount of logarithms of individual numbers.

Rule 2: Quotient Rule


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The logarithm the the quotient of numbers is the distinction of the logarithm of individual numbers.

Rule 3: strength Rule


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The logarithm of one exponential number is the exponent times the logarithm that the base.

Rule 4: Zero Rule


The logarithm of one exponential number where its basic is the very same as the base of the log amounts to the exponent.

Rule 7: Exponent of log in Rule


Raising the logarithm of a number come its base equates to the number.

Examples of just how to integrate or condensation Logarithms

This is theProduct dominance in reversebecause they space the sum of log in expressions. That method we canconvert those addition operations (plus symbols) outside into multiplication inside.


Since we have actually “condensed” or “compressed” three logarithmic expressions right into one log in expression, then that have to be our last answer.

Example 2: combine or condensation the following log expressions into a single logarithm:


The difference between logarithmic expressions indicates the Quotient Rule. I can put with each other that change x and continuous 2 within a solitary parenthesis using department operation.


Start by applying Rule 2 (Power Rule) in reverse to take treatment of the constants or numbers on the left of the logs. Remember the Power rule brings under the exponent, so the opposite direction is to put it up.

The following step is to usage the Productand Quotient rule from left come right. This is exactly how it looks when you settle it.


I can use the turning back of Power rule to place the index number on change x for the two expressions and leave the third one for now since it is already fine. Next, utilize the Product dominance to attend to the plus symbol adhered to by the Quotient ascendancy to address the individually part.

In this problem, watch out for the opportunity where you will certainly multiply and divide exponential expressions. Just a reminder, you add the exponents throughout multiplication and also subtract during division.


I suggest that girlfriend don’t skip any kind of steps. Unnecessary errors can be prevented by gift careful and also methodical in every step. Check and also recheck your occupational to make sure that friend don’t miss any important opportunity to simplify the expressions further such together combining exponential expressions through the same base.

So for this one, begin with the very first log expression by using the Power ascendancy to address that coefficient of \large1 \over 2. Next, think the the strength \large1 \over 2 together a square source operation. The square root can absolutely simplify the perfect square 81 and the y^12 since it has actually an also power.

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The steps associated are very comparable to previous problems yet there’s a “trick” the you must pay attention to. This is an interesting problem because of the consistent 3. We have to rewrite 3 in the logarithmic type such that it has a basic of 4. To build it, use dominance 5 (Identity Rule) in reverse because it renders sense the 3 = \log _4\left( 4^3 \right).