The reverse procedure of expanding logarithms is called combining or condensing logarithmic expressions right into a single quantity. Other textbooks describe this as simplifying logarithms. But, they all average the same.

You are watching: Write the expression as a single logarithm.

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

I extremely recommend that you evaluation the rules of logarithms first before looking at the worked instances belowbecause you’ll usage them in reverse.

For instance, if you go from left to appropriate of the equation climate youmust be expanding, while going from best to left then you must be condensing.



Study the summary of each ruleto obtain an intuitive knowledge of it.

Description of every Logarithm Rule


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

Rule 2: Quotient Rule


The logarithm that the quotient of number is the distinction of the logarithm of separation, personal, instance numbers.

Rule 3: power Rule


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

Rule 4: Zero Rule

The logarithm of an exponential number where its base is the very same as the base of the log equals the exponent.

Rule 7: Exponent of log in Rule

Raising the logarithm that a number come its base amounts to the number.

Examples of just how to integrate or condense Logarithms

This is theProduct preeminence 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 “condensed” or “compressed” 3 logarithmic expressions right into one log expression, climate that must be our last answer.

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

The difference in between logarithmic expressions implies the Quotient Rule. I deserve to put with each other that variable x and consistent 2 inside a solitary parenthesis using division operation.

Start by using Rule 2 (Power Rule) in turning back to take care of the constants or numbers on the left that the logs. Remember the Power rule brings under the exponent, therefore the the opposite direction is to put it up.

The next step is to use the Productand Quotient rule from left to right. This is exactly how it looks when you fix it.

I can use the reverse of Power ascendancy to location the exponents on variable x because that the 2 expressions and leave the 3rd one for now because it is currently fine. Next, make use of the Product ascendancy to deal with the add to symbol complied with by the Quotient dominance to resolve the individually part.

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

I indicate that girlfriend don’t skip any steps. Unnecessary errors can be prevented by gift careful and also methodical in every step. Check and also recheck your job-related to make certain that you don’t miss any kind of important opportunity to leveling the expressions additional such as combining exponential expressions with the same base.

So because that this one, start with the very first log expression by using the Power dominion to address that coefficient of large1 over 2. Next, think the the strength large1 over 2 as a square source operation. The square source can definitely simplify the perfect square 81 and the y^12 because it has an also power.

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The steps affiliated are very comparable to vault problems yet there’s a “trick” the you have to pay fist to. This is an amazing problem due to the fact that of the consistent 3. We have to rewrite 3 in the logarithmic type such that it has actually a basic of 4. To construct it, use ascendancy 5 (Identity Rule) in reverse because it provides sense the 3 = log _4left( 4^3 ight).