Logarithms problems | Algebra for MAT PDF Download

What are logarithms?

A logarithm is said to be an exponent. When a base 2 is given with an exponent 3, then how will you calculate 2³?

As you know  2³= 2 * 2* 2= 8,  suppose a base 2 is given with a power 8, and you are asked to find the exponent  2^? = 8 , hence the exponent that produces the value is called as Logarithm. Here 2³= 8 so the exponent 3 is nothing but the logarithm of 8 with a base value of 2. In a logarithmic way, we write it as 3 = log₂8.

In the same way 10⁴ = 10,000, it can be written as log₁₀10,000 = 4.

In a general way, the logarithmic function can be written or denoted as

Log_b x = n  or   bⁿ = x

John Napier was the person to introduce the Logarithms in the 17th century. Later it was used by many scientists, navigators, engineers, etc for performing various calculations which made it simple. In simple words, Logarithms are the inverse process of the exponentiation.

Definition of Logarithm

“The logarithm of a positive real number a with respect to base b, a positive real number not equal to 1^{[nb 1]}, is the exponent by which b must be raised to yield a”.

i.e b^y= a and it is read as “the logarithm of a to base b.”

Properties of Logarithms or Rules of Logarithms

There are four basic rules of logarithms as given below:-

1. Log_b ( mn)= log_b m + log_b In this rule the multiplication of two logarithmic values is equal to the addition of their individual logarithms.

For example- log₃ ( 2y ) = log₃ (2) + log₃ (y)

2. Log_b ( m/n)= log_b m – log_b This is called as division rule. Here the division of two logarithmic values is equal to the difference of each logarithm.

For example, log₃ ( 2/ y ) = log₃ (2) -log₃ (y)

3. Log_b ( mⁿ) = n log_b ^m

This is the exponential rule of logarithms. The logarithm of m with a rational exponent is equal to the exponent times its logarithm.

4.  Log_b m = log_a m/ log_a b    

Logarithms problems:

1. Solve log ₂ (64)= ?

Solution- since 2⁶= 2 * 2 * 2 * 2 * 2 *  2 = 64, 6 is the exponent value and log ₂ (64)= 6.

2. What is the value of log₁₀(100)= ?

Solution- Since 10 ² yields you 100, here 2 is the exponent value and the value of log₁₀(100)= 2

3. By the use of property of logarithms, solve for the value of x for log₃ x= log₃ 4+ log₃ 7

Solution- By the addition rule, log₃ 4+ log₃ 7= log ₃ (4 * 7 )

Log ₃ ( 28 ). Thus, x= 28.

4. Solve for x in log ₂ x = 5

Solution- This logarithmic function can be written In the exponential form as 2 ⁵ = x

Therefore, 2 ⁵= 2 * 2 * 2 * 2 * 2 = 32, X= 32.

Applications of Logarithms:-

Logarithms are nowadays widely used in the field of science and technology. We can even find logarithmic calculators which have made our calculations much easier. These find its applications in surveying and celestial navigation purposes. They are also used in calculations such as measuring the loudness (decibels), the intensity of the earthquake regarding Richter scale, in radioactive decay, to find the acidity (pH= -log10[H+]), etc.