Combining or Condensing Logarithms

The reverse process of expanding logarithms is called combining or condensing logarithmic expressions into a single quantity. Other textbooks refer to this as simplifying logarithms. But, they all mean the same.

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

I highly recommend that you review the rules of logarithms first before looking at the worked examples below because you’ll use them in reverse.

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

log base b of the quantity (M times N) is equal to log base b of M plus log base b of N

Rules of Logarithms

The seven rules of logarithms.

Study the description of each rule to get an intuitive understanding of it.


Description of Each Logarithm Rule

Rule1: Product Rule

log base b of (MN) = log base b of (M) + log base b of (N)

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

Rule 2: Quotient Rule

log base b of (M/N) = log base b of (M) - log base b of (N)

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

Rule 3: Power Rule

log base b of (M^k) = (k) times

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

Rule 4: Zero Rule

log base b of 1 = 0 where b>0

The logarithm of 1 with b > 0 but b \ne 1 equals zero.

Rule 5: Identity Rule

log base b of b = 1, where b > 1

The logarithm of a number that is equal to its base is just 1.

Rule 6: Log of Exponent Rule

log base b of (b^k) = k

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

Rule 7: Exponent of Log Rule

b^(log base b of k) = k

Raising the logarithm of a number to its base equals the number.


Examples of How to Combine or Condense Logarithms

Example 1: Combine or condense the following log expressions into a single logarithm:

log base 2 of (x) plus log base 2 of (y) plus log base 2 of (z)

This is the Product Rule in reverse because they are the sum of log expressions. That means we can convert those addition operations (plus symbols) outside into multiplication inside.

log base 2 of (x) plus log base 2 of (y) plus log base 2 of (z) is equal to log base 2 of (xyz)

Since we have “condensed” or “compressed” three logarithmic expressions into one log expression, then that should be our final answer.


Example 2: Combine or condense the following log expressions into a single logarithm:

log base 3 of (x) minus log base 3 of (2)

The difference between logarithmic expressions implies the Quotient Rule. I can put together that variable x and constant 2 inside a single parenthesis using division operation.

log base 3 of (x) minus log base 3 of (2) is equal to log base 3 of x over 2

Example 3: Combine or condense the following log expressions into a single logarithm:

2 times the log base 5 of (m) plus 3 times the log base 5 of (k) minus 8 times the log base 5 of (y)

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

The next step is to use the Product and Quotient rules from left to right. This is how it looks when you solve it.

2 times the log base 5 of (m) plus 3 times the log base 5 of (k) minus 8 times the log base 5 of (y) is equal to log base 5 of the quantity m squared times k cubed over y raised to the 8th power

Example 4: Combine or condense the following log expressions into a single logarithm:

5 times log base 3 of (x) plus 2 times log base 3 of 4x minus log base 3 of 8x^to the fifth power

I can apply the reverse of Power rule to place the exponents on variable x for the two expressions and leave the third one for now because it is already fine. Next, utilize the Product Rule to deal with the plus symbol followed by the Quotient Rule to address the subtraction part.

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

5 times log base 3 of (x) plus 2 times log base 3 of 4x minus log base 3 of 8x^to the fifth power is equal to log base 3 times 2x squared

Example 5: Combine or condense the following log expressions into a single logarithm:

one half of log base 7 of the quantity 81 y raised to the 12th power minus log base 7 of (3) plus log base 7 of 2 y squared

I suggest that you don’t skip any steps. Unnecessary errors can be prevented by being careful and methodical in every step. Check and recheck your work to make sure that you don’t miss any important opportunity to simplify the expressions further such as combining exponential expressions with the same base.

So for this one, start with the first log expression by applying the Power Rule to address that coefficient of \large{1 \over 2}. Next, think of the power \large{1 \over 2} as a square root operation. The square root can definitely simplify the perfect square 81 and the {y^{12}} because it has an even power.

one half of log base 7 of the quantity 81 y raised to the 12th power minus log base 7 of (3) plus log base 7 of 2 y squared is equal to log base 7 of 6 y squared

Example 6: Combine or condense the following log expressions into a single logarithm:

3 plus one half of log base 4 of (x) plus one half of log base 4 of (y)

The steps involved are very similar to previous problems but there’s a “trick” that you need to pay attention to. This is an interesting problem because of the constant 3. We have to rewrite 3 in the logarithmic form such that it has a base of 4. To construct it, use Rule 5 (Identity Rule) in reverse because it makes sense that 3 = {\log _4}\left( {{4^3}} \right).

3 plus one half of log base 4 of (x) plus one half of log base 4 of (y) is equal to log base 4 of 64 times the sqrt of xy

Practice with Worksheets


You might also be interested in:

Expanding Logarithms

Logarithm Explained

Logarithm Rules

Solving Logarithmic Equations