The logarithm of a number with a given base is the exponent to which it is necessary to raise the base in order to obtain the number. Mathematically a logarithm is represented as:

log_a x = y where a > 0 and a \neq 1

In words, we can say that the logarithm of x with base a is y.

A logarithm tells us the "number of times" a base is multiplied to get a specific number.

For example, log_2 4 =2 tells us a number of times 2 is multiplied to get the answer 4. Similarly, log_3 9 = x tells us the number of times 3 is multiplied by itself to yield the number 9.

Just like addition and subtraction are two opposite operations, logarithms and exponential functions are opposite to each other. A logarithmic function can be converted into an exponential form like this:

a ^ y = x

In words, we can say that a raised to the power y is x.

We know that when any number is raised to a power 0, the answer is 1.  In logarithmic form, we can conclude that the logarithm of a number 1 is equal to 0. Mathematically it can be written as log 1 = 0.

Remember that the base of the exponential and logarithmic function is the same. For example, the logarithmic form of the exponential function 8 ^ 2 = 64 is log _ 8 64 =2. The base of the exponential function 8 ^ 2 = 64 is 8, whereas the base of the logarithmic function log _ 8 64 =2 is also 8. Remembering this relationship will be very useful for you in understanding the logarithmic functions and will help to avoid any confusion while converting them into the exponential form or vice versa.

In this article, we will see how learn graphing logarithms, evaluating simple logarithms, natural and common logarithms and applying logarithm rules to solve complex logarithmic functions.


Graphing Logarithms

If we take the inverse of any function and graph it on a coordinate axis, then the graph of the inverse will be a reflected form of the graph of the original function. Since logarithmic functions are inverse of exponential functions, therefore the graph of the logarithmic function will be a reflection of the graph of the exponential function. Lets us consider an exponential function y = 4 ^ x. Graphically, it will be represented as follows:

Exponential function graph

The logarithmic form of the function y = 4 ^ x is log _{4} y = x. The graph of the logarithmic function will be reflected about the line y=x.

logarithmic function graph


It is pertinent to note that the graph of an exponential function is passing through the points (0,1) whereas the graph of a logarithmic function is passing through the points (1,0). The x values in the exponential function become the y values in the corresponding logarithmic function and y values in the exponential function become x values in the corresponding logarithmic function. This makes construction of a table of values easier while graphing logarithmic functions because you can simply reverse the values of corresponding exponential functions to get the tables of values for the logarithmic function.



How to Evaluate Logarithms

Most calculators will only have an option to calculate the values of common logarithms. When you have to evaluate a logarithmic function in your exam, you are expected to follow a due procedure to write an answer in the form of a fraction or a number. The answer to a logarithmic function can be positive, negative or a fraction.

Let us solve some examples related to logarithms.


Example 1

Evaluate log_\frac{1}{2} 0.25 = y


Convert the above log function into an exponential form:

(\frac{1}{2}) ^ y = 0.25

We know that \frac{1}{2} = 0.5. When we take square of 0.5, we get the answer 0.25. Hence, 0.5 should be multiplied 2 times to get 0.25. It can be written as:

\frac{1}{2} ^ y = \frac{1}{2} ^ 2

y = 2


Example 2

Evaluate log _{\sqrt{5}} (125) = y


Convert the above logarithmic function into an exponential form:

\sqrt{5} ^ y = 125

When we multiply 5 three times we get 125. But in the above question, we have got \sqrt 5 instead of 5. We know that 5 = \sqrt 5 \cdot \sqrt5. Hence, when we will multiply \sqrt 5 six times, we will get 125. It can be written as:

5 ^ {\frac{1}{2} \cdot 6} = 5 ^ 3

So, y = 6


Example 3

Evaluate log 0.001 = y


The above logarithmic function has no base. When there is no base given, then we assume that it is a common logarithmic function with base ten. Hence, the above log function has base ten which can be written as:

log_{10} 0.001 = y

Writing the above function exponentially, we will get the following algebraic expression:

10 ^ y =0.001

We know that 10 ^ 3 = 1000 and 10 ^ {-3} = \frac{1}{1000}. Since \frac{1}{1000} = 0.001, hence the value of y is -3.

10 ^ {-3} = 0.001

y = -3


Example 4

Evaluate ln \frac{1}{e ^ 5} = y


Write the above function into an exponential form like this:

e ^ y = \frac{1}{e ^ 5}

We know that when we take reciprocal of a fractional number, the sign of the power is reversed.  Therefore, e ^ y = e ^ {-5}. Hence, y = -5.


Example 5

Evaluate log _ {\sqrt {3}} (\frac {1}{81}) ^ \frac {1}{5}


Writing the above function in exponential form, we will get the following expression:

\sqrt{3} ^ y = (\frac {1}{81}) ^ \frac{1}{5}

When 3 is multiplied 4 times, we get 81.

\sqrt{3} ^ y = (\frac {1}{3 ^{ 4}}) ^ \frac{1}{5}

We know that we can write \frac {1}{3 ^ {4}} as 3 ^ {-4}, so the expression will be like this:

\sqrt{3} ^ {\frac{1}{2}y} = 3 ^{- \frac {4}{5}}

{\frac {1}{2}y} ={- \frac{4}{5}}

To isolate y on the left hand side of the expression, multiply both sides by 2 to get the final answer:

y = {-\frac{8}{5}}


Common Logarithm

A common logarithm has base 10 and is represented as log (x). When we are not given the base of the logarithm, then we assume that it has base 10. The following table shows the answers to some of the common logarithms:

Common logarithm
(log x)
log 101
log 1002
log 10003
log 0.1-1
log 0.01-2
log 0.001-3


Natural Logarithm

The natural logarithm has a base of e (an irrational number with an approximate value of 2.718281828). It is represented by ln (x).


Solving Complex Logarithmic Functions

So far we have learned to solve simple logarithmic functions. However, sometimes we are given complex logarithms that can only be solved by applying various logarithmic rules. The summary of these logarithmic rules is presented in the following table.

Logarithm RulesMathematical Notation
Logarithm product rule
Logarithm quotient rule
Logarithm power rule
Logarithm base switch rule
Logarithm root rule
Logarithm change of base rule

Example 1

Evaluate log _ 3 729 - 2 \cdot  log _ 3 3 +log _ 5 125 + log _ 5 5


As you can see that this is a complex example which involves many logarithmic factors in a single expression. We will apply logarithmic rules to solve this expression. There are two logs with the same base 3 and other two log with the same base 5. Apply logarithm quotient rule to the first two factors of the expression because a negative sign is involved. Similarly, apply logarithm product rule to the last two factors of the expression because they have a positive sign between them.

=\frac {log_{3}729}{2 log_{3} 3} + log_ 5 (125 \cdot 5)

According to the logarithm power rule, log_b x ^m = m log _b x, so we can write the expression 2 log_{3} 3 as  log _ 3 3 ^ 2 which is equal to log _ 3 9.

=log _ {3} \frac{729}{9} + log_5 625

=log _ 3 81 +log _ 5 625

Once again we will employ logarithm power rule here. We know that log _ 3 81 = log _ 3 3 ^4 and log_5 625 = log_5 5 ^4. We can write log _ 3 3 ^4 as 4 \cdot log_3 3 and log_5 5 ^4 as 4 \cdot log_5 5.

=4 \cdot log_3 3 +4 \cdot log_5 5

Converting log _3 3 =x into exponential form we get 3 ^ x = 3. Hence, we get an answer x = 1. Similarly, log_5 5 = x can be written exponentially as 5 ^ x = 1. Once, again we get x = 1. Put these two values in the above expression to get the final answer:

= 4 \cdot 1 + 4\cdot 1



Example 2

Expand log _ 5 (125 x ^{5} y ^ {4})


Apply the product rule to break down the single factor inside parenthesis into set of individual factors.

=log _ 5 125 + log _ {5} x ^ 5 + log _ {5} y ^ 4

According to the logarithm power rule, log_b x ^m = m log _b x, so we can write the expression log _ {5} x ^ 5 as  5 \cdot log_5 (x) and log_ {5} y ^ 4 as 4 \cdot log_5 (y).

=log _ 5 125 + 5 \cdot log _ {5} x  + 4 log_ {5} y

Among all the factors, we can solve only log _ 5 125. Suppose log _ 5 125 = x. Convert it into exponential form to get the value for x.

5 ^ x = 125

Since, 5 ^ 3 =125, therefore x = 3. Put this value in the expression:

=3 + 5 \cdot log _ {5} x  + 4 \cdot log_ {5} y


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