Logarithm is an exponent measure. An equation in exponential form can be equivalently written in logarithmic form. For example, we have the exponential statement, 2
^{3} = 8. This can be written in logarithmic form as log_{2} 8 = 3. Here, 2 is the base in both the equations and only the positions of 3 and 8 are switched. Equations in one form are often written in another form and solved. As functions, exponential and logarithmic functions are inverses of each other. Corresponding to properties of exponents, we also have properties of logarithm. |

y = log

_{a}x <=> x = a

^{y}.

In other words, the equations y = log

_{a}x and x = a

^{y}are equivalent equations.

We can understand y as the power to which a is raised to get x.

y = log

_{a}a

^{y}.

The function f(x) defined as f(x) = log

_{a}x is called the logarithmic function with base a.

The logarithms is defined only for positive numbers. The base of a logarithm is also positive and never equal to 1.As a

^{0}= 1, log

_{a}1 = 0. That is, logarithm of 1 to any base = 0.

Also, log 0 to any base is written as - $\infty$. Base 10 logarithms are known as common logarithm, as they are easy to manipulate and used widely in calculations.

The base of a logarithm by default is 10. That is, for the expression log x, it is understood that the base is 10.

The common logarithms of powers of 10 are integers.

log 10 = log

_{10}10

^{1}= 1

log 100 = log

_{10}10

^{2}= 2

log 10,000 = log

_{10}10

^{4}= 4

log 1 = log

_{10}10

^{0}= 0

log 0.1 = log

_{10}10

^{-1}= -1

log 0.001 = log

_{10}10

^{-3}= -3

In general, using inverse property, log 10

^{x}= x and 10

^{log x}= x.

Any number can be expressed as a power of 10. The numerical calculations involving multiplication, division and exponentiation are done taking logarithms and using properties of logarithm. The output of these logarithmic manipulations are then raised to the power of 10 (anti logarithm) to get the required result.

All Scientific calculators have the 'log' function key which is used for evaluating common logarithm. The base in natural logarithm is the natural number or Euler's constant 'e'. Natural logarithm is represented by 'ln'.

ln x = log

_{e}x where, x > 0.

y = ln x <=> x = e

^{y}.

The function f(x) = ln x is the inverse function of the natural exponential function y = e

^{x}.

In general, ln e

^{x}= x and e

^{ln x}= x (By Inverse Property)

The common rules applicable to logarithms are also known as properties or laws of logarithms. The first three laws equivalent to laws of exponents are used for simplifying exponents.

**Product Rule:**Logarithm of a product is equal to the sum of the logarithms of the factors.

log

_{b}(xy) = log

_{b}x + log

_{b}y

log 400 = log 100 + log 4 = 2 + log 4

**Quotient Rule:**Logarithm of a quotient is the difference of the logarithms of the two quantities that make the quotient.

log

_{b}(x/y) = log

_{b}x - log

_{b}y

log

_{2}(8/3) = log

_{2}8 - log

_{2}3 = log

_{2}2

^{3}- log

_{2}3 = 3 - log

_{2}3.

**Power Rule:**log

_{b}x

^{m}= m log

_{b}x

log 27 = log 3

^{3}= 3 log 3

**Change of Base Rule:**In addition to the above three rules, change of base rule is extensively used in simplifying and evaluating logarithmic expressions. The rule to change the base from b to a is as follows:

log

_{b}x = $\frac{\log_{a}x}{\log_{a}b}$

log

_{2}5 = $\frac{\log _{10}5}{\log _{10}2}$ = $\frac{\log 5}{\log 2}$

The product, quotient and power rules are used in expanding logarithms.

### Solved Example

**Question:**Expand the logarithm log

_{2}$\frac{9(x - 2)}{x^{2}}$

**Solution:**

log

= log

= log

= 2 log

_{2}$\frac{9(x - 2)}{x^{2}}$ = log_{2}(9(x - 2)) - log_{2}x^{2}(Quotient Rule)= log

_{2}9 + log_{2}(x - 2) - log_{2}x^{2}(Product Rule)= log

_{2}3^{2}+ log_{2}(x - 2) - log_{2}x^{2}= 2 log

_{2}3 + log_{2}(x - 2) - 2 log_{2}x (Power Rule)

**Examples:**- log 2 + log 5 = log (5 . 2) = log 10 = log
_{10}10^{1}= 1. - 3 ln x + ln 4 = ln x
^{3}+ ln 4 = ln (x^{3}. 4) = ln (4x^{3})

- log 16 - 2 log 2 = log 16 - log 2
^{2}= log 16 - log 4 = log (16/4) = log 4 - 3 ln(x) - ln(y) = ln x
^{3}- ln y = ln (x^{3}/y)

### Solved Examples

**Question 1:**Solve log

_{2}(x

^{2}+ 5x) = log

_{2}6

**Solution:**

x

x

(x + 6) (x - 1) = 0

=> x = - 6 and x = 1.

Both the values satisfy the given equation. Hence, the two solutions are x = - 6, 1.

^{2}+ 5x = 6 One - to - One Property of Logarithmsx

^{2}+ 5x - 6 = 0(x + 6) (x - 1) = 0

=> x = - 6 and x = 1.

Both the values satisfy the given equation. Hence, the two solutions are x = - 6, 1.

**Question 2:**Solve log

_{4}(7x + 2) = 2

**Solution:**

7x + 2 = 4

7x + 2 = 16

7x = 14

x = 4

^{2}(By definition of logarithm)7x + 2 = 16

7x = 14

x = 4

### Practice Problems

**Question 1:**Solve log (10x) + log (x - 3) = 2.

**Question 2:**Simplify: log

_{4}2 + log

_{4}8

**Question 3:**Calculate log

_{7}12 using change of base rule.