# Concept: Logarithms

CONTENTS

**INTRODUCTION**

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 (∵ b^{0} = 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 = 3^{4},

∴ 4 = log_{3} 81

- 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.

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

The logarithm of 1 to any base is always 0.

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

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

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.

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

**log**_{bn}m = $\frac{{\mathrm{log}}_{b}m}{n}$Logarithms expressed in one base can be converted to logarithms expressed in any other base.

**log**_{b}m = $\frac{1}{{\mathrm{log}}_{m}b}$**a**^{logb m}= m^{logb a}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

i.e., **log _{b} 1 = 0 ** (provided that b is positive and ≠ 1)

**Explanation**:

For any real number b, b^{0} is defined as 1.

Hence, log_{b} 1 = 0

i.e., **log _{b} b = 1 ** (provided that b is positive and ≠ 1)

**Explanation**:

For any real number b, b^{1} = b
Hence, log_{b} b = 1

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

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 log_{10} 2 = 0.301, what is the value of log_{10} 5?

**Solution**:

log_{10} 5 = log_{10} 10/2

⇒ log_{10} 5 = log_{10} 10 - log_{10} 2

⇒ log_{10} 5 = 1 - 0.301 = 0.699

i.e., **log _{b} m^{n} = n × log_{b} m** (provided that b is positive and ≠ 1)

**Explanation**:

log_{b} m^{n} = 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 log_{10} 2 = 0.301, what is the value of log_{10} 16?

**Solution**:

log_{10} 16 = log_{10} 2^{4}

⇒ log_{10} 16 = 4 × log_{10} 2

⇒ log_{10} 16 = 4 × 0.301 = 1.204

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

**Example** : If log_{2} 3 = a, what is the value of log_{0.125} 27 in terms of a?

**Solution**:

log_{0.125}27 = log_{1/8} 3^{3}

⇒ log_{0.125} 27 = 3 × log_{2-3} 3

⇒ log_{0.125} 27 = $\frac{3}{-3}$ × log_{2} 3

⇒ log_{0.125} 27 = -1 × a = -a

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

**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 = $\frac{{\mathrm{log}}_{a}m}{{\mathrm{log}}_{a}b}$

**Example** : Simplify $\frac{{\mathrm{log}}_{5}\left(8\right)}{{\mathrm{log}}_{5}\left(16\right)}$

**Solution**:

$\frac{{\mathrm{log}}_{5}\left(8\right)}{{\mathrm{log}}_{5}\left(16\right)}$ = log_{16} 8

= log_{24} 2^{3}

= 3/4 × log_{2} 2 = 3/4 = 0.75

**Explanation**:

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

**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, log_{10} 9100

= log_{10} (10^{3} × 9.1)

≈ log_{10} 10^{} + log_{10} 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:

∵ log_{10} 9100 = 3.959,

∴ 9100 = 10^{3.959} = 10^{3 + 0.959} = 10^{3} × 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, log_{10} 0.091 = log_{10} (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