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Logarithms - Laws of Operations

(Simplifying Logarithmic Expressions)

In this section we learn the rules for operations with logarithms, which are commonly called the laws of logarithms.

These rules will allow us to simplify logarithmic expressions, those are expressions involving logarithms.

For instance, by the end of this section, we'll know how to show that the expression: \[3.log_2(3)-log_2(9)+log_2(5)\] can be simplified and written: \[log_2(15)\]

To do this we learn three rules:

  • the addition rule for logarithms
  • the subtraction rule for logarithms
  • the power rule for logarithms
we'll state the rule and see a detailed tutorial for each of these rules as well as learn a few "must-know tricks" along the way.

Let's get started.

Addition Law

When adding two logarithms, in the same base \(b\), the following simplification can always be made: \[log_b(a)+log_b(c) = log_b(a\times c)\]

Example

The expression: \[log_3(5)+log_3(8)\] can be simplied and written: \[\begin{aligned} log_3(5)+log_3(8) & = log_3(5\times 8) \\ & = log_3(40) \end{aligned}\]

Tutorial 1: Addition Rule for Logarithms

Exercise 1

Simplify each of the following as much as possible:

  1. \(log_3(5)+log_3(2)\)
  2. \(log_7(8)+log_7(10)\)
  3. \(log_5(6)+log_5(2)+log_5(3)\)
  4. \(log_b \begin{pmatrix} p \end{pmatrix} +log_b\begin{pmatrix} q \end{pmatrix}\)
  5. \(log_b \begin{pmatrix} p \end{pmatrix} +log_b\begin{pmatrix} p^2 \end{pmatrix}\)

Answers Without Working

We find the following results:

  1. \(log_3(5)+log_3(2) = log_3(10)\)
  2. \(log_7(8)+log_7(10) = log_7(80)\)
  3. \(log_5(6)+log_5(2)+log_5(3) = log_5(36)\)
  4. \(log_b \begin{pmatrix} p \end{pmatrix} +log_b\begin{pmatrix} q \end{pmatrix} = log_b \begin{pmatrix} pq \end{pmatrix}\)
  5. \(log_b \begin{pmatrix} p \end{pmatrix} +log_b\begin{pmatrix} p^2 \end{pmatrix} = log_b \begin{pmatrix} p^3 \end{pmatrix} \)

Subtraction Law

When subtracting with logarithms, in the same base \(b\), the following simplification can always be made: \[log_b(a)-log_b(c) = log_b \begin{pmatrix} \frac{a}{c}\end{pmatrix}\]

Example

The expression: \[log_2(50)-log_2(10)\] can be simplied and written: \[\begin{aligned} log_2(50)-log_2(10) & = log_2\begin{pmatrix}\frac{50}{10}\end{pmatrix} \\ & = log_2(5) \end{aligned}\]

Tutorial 2: Subtraction Rule for Logarithms

Exercise 2

Simplify each of the following as much as possible:

  1. \(log_3(8)-log_3(4)\)
  2. \(log_{10}(24)-log_{10}(6)\)
  3. \(log_b\begin{pmatrix} p^3 \end{pmatrix} - log_b\begin{pmatrix} p^2 \end{pmatrix} \)
  4. \(log_5(64)-log_5(4)-log_5(2)\)
  5. \(log_b\begin{pmatrix} p \end{pmatrix} + log_b\begin{pmatrix} p^3 \end{pmatrix} - log_b\begin{pmatrix} p^2 \end{pmatrix}\)

Answers Without Working

We find the following results:

  1. \(log_3(8)-log_3(4) = log_3(2)\)
  2. \(log_{10}(24)-log_{10}(6) = log_{10}(4)\)
  3. \(log_b\begin{pmatrix} p^3 \end{pmatrix} - log_b\begin{pmatrix} p^2 \end{pmatrix} = log_b\begin{pmatrix} p \end{pmatrix}\)
  4. \(log_5(64)-log_5(4)-log_5(2) = log_5(8)\)
  5. \(log_b\begin{pmatrix} p \end{pmatrix} + log_b\begin{pmatrix} p^3 \end{pmatrix} - log_b\begin{pmatrix} p^2 \end{pmatrix} = log_b\begin{pmatrix} p^2 \end{pmatrix}\)

Power Rule for Logarithms (multiplication by a scalar)

When a logarithm, base \(b\) is multiplied by a scalar, \(x\), the following simplification can always be made: \[x.log_b(a)= log_b \begin{pmatrix} a^x\end{pmatrix}\]

Example

The following expression: \[4.log_6(2)\] can be simplified as: \[\begin{aligned} 4.log_6(2) & = log_6\begin{pmatrix}2^4\end{pmatrix}\\ & = log_6(16) \end{aligned}\]

Tutorial 3: Power Rule for Logarithms

Exercise 3

Simplify each of the following as much as possible:

  1. \(3. log_2(5)\)
  2. \(2. log_5(3) + 3. log_5(2)\)
  3. \(6. log_b\begin{pmatrix}p \end{pmatrix} - log_b\begin{pmatrix}p^2 \end{pmatrix}\)
  4. \(2.log_4(a)-5.log_4(b)\)
  5. \(5.log_2(m)+2.log_2 \begin{pmatrix}n^3\end{pmatrix}\)

Answers Without Working

Simplify each of the following as much as possible:

  1. \(3. log_2(5) = log_5\begin{pmatrix} 125\end{pmatrix}\)
  2. \(2. log_5(3) + 3. log_5(2) = log_5 \begin{pmatrix} 72 \end{pmatrix}\)
  3. \(6. log_b\begin{pmatrix}p \end{pmatrix} - log_b\begin{pmatrix}p^2 \end{pmatrix} = log_b\begin{pmatrix}p^4 \end{pmatrix}\)
  4. \(2.log_4(a)-5.log_4(b) = log_4\begin{pmatrix} \frac{a^2}{b^5}\end{pmatrix}\)
  5. \(5.log_2(m)+2.log_2 \begin{pmatrix}n^3\end{pmatrix} = log_2\begin{pmatrix} m^5.n^6\end{pmatrix}\)

Some "must-know" results & tricks

We now learn how to deal with numbers being added or subtracted to a logarithm. In particular, we learn how to write any number as a logarithm.

For instance we may be required to simplify the expression: \[3 + log_3\begin{pmatrix}5\end{pmatrix}\]

Writing any number as a Logarithm

Any number \(k\) can be written as a logarithm in any base \(b\) using the following result: \[k = log_b\begin{pmatrix}b^k\end{pmatrix}\]

Example

Say we wish to simplify the expression: \[3+log_2(5)\] Then the trick is to write \(3\) as a logarithm in base \(2\) and then use the addition rule to simplify.

Using the result, written above, we can state: \[3 = log_2\begin{pmatrix}2^3\end{pmatrix}\] And so we can rewrite and simplify the expression as follows: \[\begin{aligned} 3+log_2(5) & =log_2\begin{pmatrix}2^3 \end{pmatrix} + log_2\begin{pmatrix} 5 \end{pmatrix} \\ & = log_2 \begin{pmatrix}8 \end{pmatrix} + log_2 \begin{pmatrix} 5 \end{pmatrix} \\ & = log_2 \begin{pmatrix}8\times 5 \end{pmatrix} \\ 3+log_2(5) & = log_2 \begin{pmatrix}40 \end{pmatrix} \end{aligned}\] This technique is further illustrated in the tutorial below.

Tutorial: Numbers Added To, or Subtracted From, a Logarithm

Exercise 4

Simplify each of the following as much as possible:

  1. \(2+log_3(4)\)
  2. \(4 - log_2(3)\)
  3. \(1+2.log_3(5)\)
  4. \(1+3.log_5(2)\)
  5. \(3 - log_{10}(5)\)

Answers Without Working

We find the following:

  1. \(2+log_3(4) = log_3(36)\)
  2. \(4 - log_2(3) = log_2\begin{pmatrix} \frac{16}{3}\end{pmatrix}\)
  3. \(1+2.log_3(5) = log_3(75)\)
  4. \(1+3.log_5(2) = log_5(40)\)
  5. \(3 - log_{10}(5) = log_{10}(200)\)


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