What is logarithm

Logarithmic and exponential functions are inverses of each other. For example, a ^ n = b is an exponential function. In logarithmic form, it can be written as:

log_a(b) = n

A logarithm of a number with the log base e is known as a natural logarithm. It is expressed mathematically as ln x or log _e (x). The value of e is equal to 2.71828... The logarithm of a number to the base ten is expressed as log _ {10} (x) and is known as a common logarithm.

Basic Properties of Logarithms

Some of the basic properties of the logarithmic function are as follows:

  • The base of the logarithmic function should be greater than 0 and not equal to 1, i.e. in log_a x, a>0 and a\neq 1.
  • The base of a logarithm cannot be negative which means it should always be positive
  • The logarithm of zero does not exist
  • The logarithm of 1 is zero. It can be expressed as log _ a 1 = 0, where a is the base of the logarithm
  • The logarithm with the same base and number is 1. For example, the logarithm of the number b with base b is one. Mathematically it can be expressed as log_b (b) = 1. For example, we can write the logarithmic function log_3 3 = x in exponential form as 3^x = 3. Since, 3 raised to the power 1 is equal to 3, so x = 3.
  • If logarithm is in exponential form and base and the number of the logarithm is the same, then it is equal to the value of the exponent. Mathematically it can be expressed as log_a a^n = n.

 

Logarithm Rules

When the question involves the term "expand", you need to apply logarithmic rules to simplify the expression and solve for values. Don't find logarithms using a calculator because you are expected to write the answers in whole numbers or fractional form, whereas the calculator will give you a decimal number. Just like rules of exponents, there are different logarithmic rules. These logarithmic rules along with the relevant examples are explained below:

1  Logarithm Product Rule

The logarithm of a product of two numbers is equal to the sum of the logarithm of the numbers. It can be expressed mathematically as:

log_a (m.n) = log_a m + log_a n

 

Examples

1   Expand log_2 (4 \cdot 8)

According to the logarithm product rule log_2 (4 \cdot 8) = log_2 4 + log_2 8

Suppose, log_2 4 =x and log_2 8 = y.  Since, logarithm is an inverse function of an exponential equation, so we will rewrite these two logarithms into exponential equations like this:

log_2 4 = x      2 ^ x = 4

log_2 8 = y     2 ^ y = 8

If 2 is raised to a power 2, the answer is 4, i.e. 2 ^ 2 = 4 . Similarly, if 2 is raised the power 3, the answer is 8, i.e.  2 ^ 3 = 8. Hence, the values for x and y are 2 and 3 respectively. Hence, the final answer is:

log_2 (4 \cdot 8) = log_2 4 + log_2 8

= 2 + 3 = 5

 

2   Expand log_5 (125 \cdot 625 )

According to the logarithm product rule log_5 (125 \cdot 625) = log_5 125 + log_5 625

Suppose, log_5 125 =x and log_5 625 = y. Write these two logarithmic functions in exponential forms like this:

log_5 125 = x      5 ^ x = 125

log_5 625 = y     5 ^ y = 625

If 5 is raised to the power 3, the answer is 125, i.e. 5 ^ 3 = 125 . Similarly, if 5 is raised the power 4, the answer is 625, i.e.  5 ^ 4 = 625. Hence, the values for x and y are 3 and 4 respectively. Hence, the final answer is:

log_5 (125 \cdot 625) = log_5 125 + log_5 625

= 3 + 4 = 7

2  Logarithm Quotient Rule

The logarithm of division of m and n is equal to the difference of logarithm of m and logarithm of n. Remember that the rule is true for the logarithm of numerator - logarithm of denominator, not the other way around. Mathematically, it can be expressed as:

log _b (\frac{m}{n}) = log_b m - log_b n

Examples

1 Expand log_2 (\frac{8}{4})

According to the logarithm quotient rule, log_2 (\frac{8}{4}) = log_2 8 - log_2 4. Suppose log_2 8 = x and log_2 4 = y.  Write these two logarithmic functions in exponential forms like this:

log_2 8 = x        2 ^ x = 8

log_2 4 = y        2 ^ y = 4

We know that when 2 raised to the power 3 is equal to 8, i.e. 2 ^ 3 = 8 and 2 raised to the power 2 is equal to 4, i.e. 2 ^ 2 = 4. Hence, the final answer will be:

log_2 (\frac{8}{4}) = log_2 8 - log_2 4

3 - 2 = 1

 

2  Expand log_2 (\frac{64}{32})

According to the logarithm quotient rule, log_2 (\frac{64}{32}) = log_2 64 - log_2 32. Suppose log_2 64 = x and log_2 32 = y.  Write these two logarithmic functions in exponential forms like this:

log_2 64 = x        2 ^ x = 64

log_2 32 = y        2 ^ y = 32

We know that  2 raised to the power 6 is equal to 64, i.e. 2 ^ 6 = 64 and 2 raised to the power 5 is equal to 32, i.e. 2 ^ 5= 32. Hence, the final answer will be:

log_2 (\frac{64}{32}) = log_2 64 - log_2 32

6 - 5 = 1

Remember that the logarithm product and quotient rules are only applicable to the function having the same base. For example, from the above examples you can conclude that log_a p - log_a q can be written as a single logarithmic function as log_a \frac{p}{q}. The expressions log_a p - log_a q are combined to form a single expression log_a \frac{p}{q} because they had a common base a. We cannot combine  log_a p - log_b q into a single expression because both the logarithmic functions have different bases a and b. The same rule applies to the logarithm product rule.

 

3 Logarithm Power Rule

The logarithm of the power is equal to the product of the power or exponent  and the logarithm of the number. It can be expressed mathematically as:

log_b (x ^ m) = m log_b x

Examples

1  Expand log_ 2 (8 ^ 4)

According to the logarithm power rule, log_ 2 (8 ^ 4) = 4 log _ 2 8

Suppose log_2 8 = x. First, we will convert log_2 8 into exponential form like this:

2 ^ x = 8

We know that 2 raised to the power 3 is equal to 8, i.e. 2 ^ 3 = 8, so log_ 2 (8 ^ 4) = 4 \cdot 3 = 12.

 

2 Expand log_ 3 (9 ^ 5)

According to the logarithm power rule, log_ 3 (9 ^ 5) = 5 log _ 3 9

Suppose log_3 9 = x. First, we will convert log_3 9 into exponential form like this:

3 ^ x = 9

We know that 3 raised to the power 2 is equal to 9, i.e. 3 ^ 2 = 9, so log_ 3 (9 ^ 5) = 5 \cdot 2 = 10.

 

4   Logarithm Base Switch Rule

The base a of logarithmic function b is equal to the reciprocal of logarithm a with base b. It can be expressed mathematically as:

log_a (b) = \frac {1}{log_b (a)}

Examples

1 Prove log_2 32 = \frac {1}{log_{32} 2}

Let us solve left hand side of the above expression first.

Suppose log_2 32 = x.

By converting the above log form into the exponential notation we get the following expression:

2 ^ x = 32

Since, 2 to the power 5 is equal to 32, i.e. 2 ^ 5 = 32, so log_2 32 = 5

Now, we will solve right hand side of the above expression \frac {1}{log_{32} 2}

Suppose log_{32} 2 = x. When we will write it in exponential form, we will get the following expression:

32 ^ x = 2

We know that 2 raised to the power 5 is equal to 32. Hence, we can say that 32 raised to the power \frac{1}{5} is equal to 2. Hence, \frac {1}{log_{32} 2} = \frac {1} {1/5} which is equal to 5.

 

2  Prove log_5 {125} = \frac {1}{log_{125} 5}

Lets solve left hand side of the above expression first.

Suppose log_5 125 = x.

Writing it in exponential notation we get the following expression:

5 ^ x = 125

Since, 5 to the power 3 is equal to 125, i.e. 5 ^ 3 = 125, so log_5 125 = 3

Now, we will solve right hand side of the above expression \frac {1}{log_{125} 5}

Suppose log_{125} 5 = x. When we will write it in exponential form, we will get the following expression:

125 ^ x = 5

We know that 5 raised to the power 3 is equal to 125. Hence, we can say that 125 raised to the power \frac{1}{3} is equal to 5. Hence, \frac {1}{log_{125} 5} = \frac {1} {1/3} which is equal to 3. Hence, this logarithmic rule is proved because left hand side is equal to the right hand side.

 

5  Logarithm Root Rule

The logarithm of a root is equal to the product between the logarithm of the radicand and the index of the root. It can be expressed mathematically as:

log_a (\sqrt x) = \frac {1}{n} log_a x

 Examples

1 Expand log_2 (\sqrt (8))

According to the logarithm root rule, log_2 (\sqrt (8)) = \frac {1}{2} log_2 8. Suppose log_2 8 = x. Convert this logarithmic expression into the exponential form like this:

2 ^ x = 8

We know that 2 raised to the power 3 is equal to 8, i.e. 2 ^ 3 = 8.

log_2 (\sqrt (8)) = \frac {1}{2} log_2 8

log _2 (\sqrt (8)) = \frac {1}{2} 3

=\frac{3}{2}

2 Expand log_ 3 (\sqrt (9))

According to the logarithm root rule, log_3 (\sqrt (9)) = \frac {1}{2} log_3 9. Suppose log_3 9 = x. Convert this logarithmic expression into the exponential form like this:

3 ^ x = 9

We know that 3 raised to the power 2 is equal to 9, i.e. 3 ^ 2 = 9. It means that log_3 9 = 2 .

log_3 (\sqrt (9)) = \frac {1}{2} log_3 9

log _2 (\sqrt (8)) = \frac {1}{2} \cdot 2

=1

6 Change of Base

This logarithmic rule can be described mathematically as:

log_a x = \frac {log_b x}{log_b a}

Examples

1   Prove log_2 4 = \frac {log_4 4}{log_4 2}

First solve left hand side of the expression first by converting it into the exponential form:

log_2 4 =x

2 ^ x = 4

Since, 2 raised to the power 2 is equal to 4, hence the value of x is equal  to 2. Now, we will solve right hand side of the equality \frac {log_4 4}{log_4 2}. Suppose log_ 4 4 = x and log_4 2 = y. Convert these logarithmic functions into exponential forms to find their values:

4 ^ x = 4 and 4 ^ y = 2

We know that 4 raised to the power 1 is equal to 4, so x = 1. Similarly, 4 raised to the power \frac{1}{2} is equal to 2, so y = \frac{1}{2}. Put these values in the expression \frac {log_b x}{log_b a}.

\frac{1}{1/2}

=2

Hence, the logarithmic rule is proved through this example because the left hand side is equal to the right-hand side.

 

2  Prove log_3 27 = \frac {log_27 27}{log_27 3}

First solve left hand side of the expression first by converting it into the exponential form:

log_3 27 =x

3 ^ x = 27

Since 3 raised to the power 3 is equal to 27, hence the value of x is equal  to 3. Now, we will solve right hand side of the logarithmic equation \frac {log_{27} 27}{log_{27} 3}. Suppose log_ {27} 27 = x and log_{27} 3 = y. Convert these logarithmic functions into exponential forms to find their values:

3 ^ x = 27 and 27 ^ x = 3

We know that 3 raised to the power 3 is equal to 27, so x = 3. Similarly, 27 raised to the power \frac{1}{3} is equal to 3, so y = \frac{1}{3}. Put these values in the right hand side of the logarithmic equation \frac {log_{27} 27}{log_{27} 3}.

\frac{1}{1/3}

=3

Hence, the left hand side of the logarithmic equation is equal to the right-hand side, so this rule is proved.

 

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.

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