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Expand Logarithmic Expressions Using a Combination Of Logarithm Rules

Referencing Styles : APA | Pages : 1

A log is in an exponent form so when we take the log of something, we are getting back an exponent. The two equations below show the two different ways that say the same thing, the only difference is the first is an exponential equation, and the second is a logarithmic equation.

Exponential Function                         Logarithmic Function

x=by                               ⇔                              y=logbx


Example:                                                           Example:

   25=52                                                            2=log525


There are certain rules in expanding logarithms and the logarithmic rules are described as follows:

Log Rules:

1) logb (mn) = logb(m) + logb(n)

2) logb (m/n) = logb(m) – logb(n)

3) logb (mn) = n · logb(m)

In less formal terms the log rules can be expressed as

  1. Multiplication inside the log can be turned into addition outside the log and vice versa
  2. Division inside the log can be turned into subtraction outside the log and vice versa.
  3. An exponent on everything inside a log can be moved out as a multiplier and vice versa.

When dealing with the exponents the above rules are applied only if the bases are same. For example, if the expression “logd(m) + logb(n)” cannot be simplified as the bases here are “d” and “b” that are not same.

Throughout the study of algebra, we have come across many properties—such as the commutative, associative, and distributive properties. These properties help us to take a complicated expression or equation and simplify it. The same is true with logarithms. There are a number of properties that will help you simplify complex logarithmic expressions. Since logarithms are so closely related to exponential expressions, it is not surprising that the properties of logarithms are very similar to the properties of exponents. As a quick refresher, here are the exponent properties.

Logarithm of a Product

In case of exponents when we need to multiply two numbers with the same then we simply add the two exponents while in case of logarithm the logarithm is a product of the sum of the logarithms.  Thus, the logarithm of a product is shown below

Logb (MN) = logb M + logb N

Logarithm of a Quotient

To divide two numbers with the same base we need to subtract the exponents. The logarithm of the quotient is the difference of the logarithms. Thus, the logarithm of a product is shown below

Logb (M/N) = logb M - logb N

Logarithm of a Power

The remaining exponent property was power of a power:  The formula for logarithm with power is given below

Logb (Mn) = n logb M


In both the properties,   the power “n” becomes a factor.

The log rules can be used to simplify the expressions that will wither expand or solve the expression for values. Given below are some examples that explains the expansion of of logarithms.

· Expand log3(2x)

Whenever the instructor says to expand the expression it means that they have given a log expression and they want us to use the log rules to make the log apart into many separate terms, each with only one term inside a particular log. In this case, we have “2x" inside the log. As “2x” is a multiplication we can take this expression apart according to the first log rules that have already been mentioned above. Then we can turn it into addition outside the log as shown below

          log3(2x) = log3(2) + log3(x)

          Then the answer they are looking for is:

            log3(2) + log3(x)


We cannot calculate "log3 (2)" in the calculator as the answer will be incorrect and the calculator will see the expression as a number and the answer will be a decimal approximation. However, the instructor is looking for the exact form of log that is not possible to get in the calculator.

·Let us take a look at another example Expand log4( 16/x ). Here there is a division inside the log. According to the second rule of log this can be split as subtraction when it is brought outside log. Hence,

log4( 16/x) = log4(16) – log4(x)

In this case, I am using the fact that the power required on 4 to create 16 is 2; in other words, since 42 = 16, then:

log4(16) = 2

Then the original expression expands fully as:

log4( 16/x ) = 2 – log4(x)

If we take the product rule, quotient rule, and the power role together we can get the properties of logs. Sometimes we can apply more than one rule in order to expand a particular expression. For example,

logb(6x/y) =logb(6x)−logby




We can use the power rule to expand logarithmic expressions involving negative and fractional exponents. Here is an alternate proof of the quotient rule for logarithms using the fact that a reciprocal is a negative power:




We can also apply the product rule to express a sum or difference of logarithms as the logarithm of a product.

We can also condense the sums, products and differences with the same base as a single logarithm. We can use the rules of logarithm to condense the expression, but it is important to keep in mind that the logarithms must have the same base so that they can be combined.

Properties of Exponents


How to write an equivalent expression as a single logarithm given a sum, difference, or product of logarithms with the same base

1. Apply the power property first. Identify terms that are products of factors and a logarithm and rewrite each as the logarithm of a power.

  1. From left to right, apply the product and quotient properties. Rewrite sums of logarithms as the logarithm of a product and differences of logarithms as the logarithm of a quotient.


Like exponents, logarithms have properties that allow you to simplify logarithms when their inputs are a product, a quotient, or a value taken to a power. The properties of exponents and the properties of logarithms have similar forms. The table below summarizes the quotient, product and power property that shows the details of the exponents and logarithms.

 Here we can notice that how the quotient property leads to subtraction, the product property leads to addition, and the power property leads to multiplication for both exponents and logarithms.


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