Concept: Logarithms

CONTENTS

• Introduction
• Characteristic & Mantissa

 

Long before the invention of calculators or any other computing device, mathematicians used logarithms to do complex calculations.

If N = b^P, where a > 0 and a ≠ 1, then

p = log_b N

Here,
N = number
b = base
p = power

In words, p is referred to as the logarithm of N to the base b.

Here, b cannot be negative, 0 or 1 because logarithm of negative numbers and 0 is not defined and the logarithm of 1 (to the base 10) is equal to 0 (∵ b⁰ = 1).

In other words, the logarithm of any number to a given base is the power to which the base must be raised to get that number.

The reverse is also true. That is, if log_b N = p, then we can write N = b^p

Logarithm is used to express power in terms of number and the base.

For example,
Since 81 = 3⁴,
∴ 4 = log₃ 81

 

Most problems involving logarithms can be solved by applying the following laws/properties:

The logarithm of 1 to any base is always 0.

i.e., log_b ⁡1 = 0 (provided that b is positive and ≠ 1)

Explanation:
For any real number b, b⁰ is defined as 1.
Hence, log_b 1 = 0




The logarithm of a number to a base which is equal to that number is 1.

i.e., log_b b = 1 (provided that b is positive and ≠ 1)

Explanation:
For any real number b, b¹ = b Hence, log_b b = 1




1. The logarithm of the product of two numbers is equal to the sum of the logarithms of the individual numbers.

i.e., log_b ⁡(m × n) = log_b m + log_b n (provided that b is positive and ≠ 1)

Explanation:
Let log_b m = x and log_b n = y
∴ m = b^x and n = b^y
∴ m × n = b^x × b^y = b^{x + y}
∴ log_b (m × n) = x + y




The logarithm of the ratio of numbers is equal to the difference obtained when the logarithm of the denominator is subtracted from that of the numerator.

i.e., log_b ⁡$\left(\frac{m}{n}\right)$ = log_b m - log_b n (provided that b is positive and ≠ 1)

Explanation:
Let log_b m = x and log_b n = y
∴ m = b^x and n = b^y
∴ $\left(\frac{m}{n}\right)$ = b^x ÷ b^y = b^{x - y}
∴ log_b $\left(\frac{m}{n}\right)$ = x - y




Example : If log₁₀ 2 = 0.301, what is the value of log₁₀ 5?

Solution:
log₁₀ 5 = log₁₀ 10/2

⇒ log₁₀ 5 = log₁₀ 10 - log₁₀ 2

⇒ log₁₀ 5 = 1 - 0.301 = 0.699




The logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number.

i.e., log_b ⁡mⁿ = n × log_b m (provided that b is positive and ≠ 1)

Explanation:
log_b mⁿ = log_b (m × m × m × ... × m)
Using law 3 above, the RHS can be expressed as
log_b (m × m × m × ... × m) = log_b m + log_b m + log_b m + ... + log_b m = n × log_b m




Example : If log₁₀ 2 = 0.301, what is the value of log₁₀ 16?

Solution:
log₁₀ 16 = log₁₀ 2⁴

⇒ log₁₀ 16 = 4 × log₁₀ 2

⇒ log₁₀ 16 = 4 × 0.301 = 1.204




log_bⁿ ⁡m = $\frac{{\log }_{b}m}{n}$

i.e., (provided that b is positive and ≠ 1)




Example : If log₂ 3 = a, what is the value of log_0.125 27 in terms of a?

Solution:
log_0.12527 = log_1/8 3³

⇒ log_0.125 27 = 3 × log_2^-3 3

⇒ log_0.125 27 = $\frac{3}{-3}$ × log₂ 3

⇒ log_0.125 27 = -1 × a = -a




Logarithms expressed in one base can be converted to logarithms expressed in any other base.

i.e., log_b m = $\frac{{\log }_{a}m}{{\log }_{a}b}$

Explanation:
Let log_b m = x
∴ m = b^x
Let a^y = b
∴ y = log_a b
If we substitute the value of b in the expression for m, we get,
m = (a^y)^x = a^xy
∴ log^a m = xy = log_b m × log_a b
∴ log_b m = $\frac{{\log }_{a}m}{{\log }_{a}b}$




Example : Simplify $\frac{{\log }_5\left(8\right)}{{\log }_5\left(16\right)}$

Solution:
$\frac{{\log }_5\left(8\right)}{{\log }_5\left(16\right)}$  = log₁₆ 8

= log_2⁴ 2³

= 3/4 × log₂ 2 = 3/4 = 0.75




log_b m = $\frac{1}{{\log }_{m}b}$

Explanation:
In the previous law, instead of changing the base to a, change the base to m.




a^{log_b m} = m^{log_b a}




2. Given an equation, log_a M = log_b N, then

• If M = N, then a will be equal to b.
• If a = b, then M will be equal to N.



In order to compare two numbers, when the comparison between their respective logarithms is given:

• If b > 1 and log_b m > log_b n, then m > n
• If b < 1 and log_b m > log_b n, then m < n

The logarithm of a number has an integral part and a decimal part. The integral part is called the characteristic and the decimal part is called the mantissa.

For example, log₁₀ 9100
= log₁₀ (10³ × 9.1)
≈ log₁₀ 10 + log₁₀ 9.1
= 3 + 0.959

Here, 3 is the characteristic and 0.959 is the mantissa.

An important property of the characteristic (for common logarithms) is that (characteristic + 1) will give the number of digits of the integral part of the number whose logarithm you determined. In the above example, 9100 contains (3 + 1) = 4 digits.

This is explained as follows:
∵ log₁₀ 9100 = 3.959,
∴ 9100 = 10^3.959 = 10^{3 + 0.959} = 10³ × 10^0.959
= 10^x × 10^y

Since y will always be < 1, 10^y will always be < 10. Hence, the number of digits in the number will be equal to the number of digits in 10^x, which in turn is equal to (x + 1).

Also, the characteristic of a logarithm could be either positive or negative; however, the mantissa should always be positive.

For example, log₁₀ 0.091 = log₁₀ (10^-2 × 9.1)
≈ –2 + 0.959 = $\overline{2}$.959

If the calculated value of the logarithm of some number is negative (usually, calculators generate values containing a negative mantissa), then we make the mantissa positive as follows:
–1.041 = –(1 + 0.041) = –1 – 0.041
= (–1 – 1) + (–0.041 + 1)
= –2 + 0.959
= $\overline{2}$.959