# Concept: Logarithms

INTRODUCTION

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

If N = bP, where a > 0 and a ≠ 1, then

p = logb 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 logb N = p, then we can write N = bp

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

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

SOME IMPORTANT RESULTS
• Logarithms are defined only for positive numbers. It is not defined for zero or negative numbers.
• Logarithms can be expressed in any base.
• Logarithms expressed in one base can be converted to logarithms expressed in any other base.
• Logarithms are generally expressed to the base 10. These are called common logarithms.

LAWS & PROPERTIES OF LOGS

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

1. The logarithm of 1 to any base is always 0.

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

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

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

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

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

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

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

Explanation:
Let logb m = x and logb n = y
∴ m = bx and n = by
∴ m × n = bx × by = bx + y
∴ logb (m × n) = x + y

7. 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.

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

Explanation:
Let logb m = x and logb n = y
∴ m = bx and n = by
$\left(\frac{m}{n}\right)$ = bx ÷ by = bx - y
∴ logb $\left(\frac{m}{n}\right)$ = x - y

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

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

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

11. logbn ⁡m = $\frac{{\mathrm{log}}_{b}m}{n}$

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

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

14. i.e., logb m = $\frac{{\mathrm{log}}_{a}m}{{\mathrm{log}}_{a}b}$

Explanation:
Let logb m = x
∴ m = bx
Let ay = b
∴ y = loga b
If we substitute the value of b in the expression for m, we get,
m = (ay)x = axy
∴ loga m = xy = logb m × loga b
∴ logb m = $\frac{{\mathrm{log}}_{a}m}{{\mathrm{log}}_{a}b}$

15. logb m = $\frac{1}{{\mathrm{log}}_{m}b}$

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

17. alogb m = mlogb a

18. Given an equation, loga M = logb N, then

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

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

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

CHARACTERISTIC AND MANTISSA

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 = 103.959 = 103 + 0.959 = 103 × 100.959
= 10x × 10y

Since y will always be < 1, 10y will always be < 10. Hence, the number of digits in the number will be equal to the number of digits in 10x, 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

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