Logarithm:

Earlier we have studied exponents which covers every part of mathematics. Now it’s contained to know that logarithm is the inverse function of exponentiation. In this article, we will read about the log formula. Logarithm is denoted as the repeated multiplication of the same factor. As any number as 81 can be factorized of same factors 3 as 3 x 3 x 3 x 3 which is 4 times multiplication of 3. Now we can say that log of 81 for factor (base) 3 is 4.

i.e. log₃(81) = 4 here, 3 is base.

Points:

• We can identify the logarithmic value of positive numbers only.
• log(0) = undefined
• The logarithmic value of negative numbers is imaginary. There is no real value of log function for negative numbers.

Log Formula:

In all these rules a>1, m, n, k be any real number but m, n must be positive. Then log formula will be valid.

Terms related to log formula:

1. Common logarithm: The log which has base of 10 is known as common log.
   Example: log(24) = log₁₀(24)
2. Natural logarithm: The log which has base of e (i.e. Euler,s number has a value of 2.71) is known as natural log. It is denoted by “ln”.
   Example: ln(24) = log_e(24)
3. Negative logarithms: It has a simple meaning that how many times we divide a number.
   Example: log₈(0.125) = -1, as 1/8 = 0.125

Log formula Examples:

1. Solve the sum of log₁₀(25) + log₁₀(4).

Solution:
we know that log_a(m.n) = log_a m + log_a n
so, log₁₀(25) + log₁₀(4)

= log₁₀(100)

= log₁₀(10)²
= 2 log₁₀(10) = 2

2. Solve log₅500 −2 log₅2.

Solution:
we can write log₅500 −2 log₅2 as log₅500 −2 log₅2

= log₅500 − log₅ (2)²

= log₅500 − log₅ (4)

As we know that log_a m – log_a n = log_a( \(\frac{m}{n}\) )

= log₅( \(\frac{500}{4}\) )

thus, log₅(125)

= log₅(5)³ = 3