# 9. Logarithm

**Definition:** A number \(x\) is called the logarithm of a number
\(y\) to the base \(b\) if \(b^x = N\) where \(b > 0, b\neq 1,
y > 0\)

Mathematically, it is represented by the equation \(\log_b y = x\) or \(b^x = y\)

**Notes:**

The conditions \(b>0, b\neq 1\) and \(N>0\) are necessary in the definition of logarithm.

When \(b = 1\) suppose logarithm is defined, and we have to find the value of \(\log_1 y.\) Let

\(\log_1 y = x\Rightarrow 1^x = y \Rightarrow 1 = y\)

If \(\log_1 2\) is defined then \(1 = 2.\) So we see that \(b = 1\) leads to meaningless result. Similarly, it is true for \(b \neq 0\)

Similarly if \(y < 0,\) then \(b^x = y\) which is meaningless as L.H.S. is positive while R.H.S. is negative

Let the condition to be true when \(b = 0.\) Thus, \(0^x = N\) or \(0 = N\) i.e. if \(=log_0 2\) is defined will mean that \(0 = 2\) Hence our assumption leads to failure.

No number can have two different logarithms to a given base. Assume that a number \(N\) has two different logarithms \(x\) and \(y\) with base \(N\). Thenm

\(\log_b N = x\) and \(\log_b N = y\)

\(N = b^x\) and \(b^y = N\)

\(\Rightarrow b^x = b^y \Rightarrow x = y\)

When the number or base is negative the value of logarithm comes out to be a complex number with non-zero imaginary part.

Let \(\log_e (-5) = x\)

\(\log_e 5.e^{i\pi} = x\) (recall from complex numbers that \(e^{i\pi} = -1\))

\(x = \log_e 5 + i\pi\)

## 9.1. Important Results

\(\log_b 1 = 0\)

**Proof:**Let \(\log_b 1 = x\Rightarrow b^x = 1 \Rightarrow x = 0\)\(=log_b b = 1\)

**Proof:**Let \(\log_b b = x\Rightarrow b^x = b\Rightarrow x = 1\)\(b^{\log_b N} = N\)

**Proof:**Let \(\log_b N = x \Rightarrow b^x = N\)\(b^{\log_b N} = N [\because x = \log_b N]\)

## 9.2. Important Formulae

\(\log_b (x.y) = \log_b x + \log_b y, (x > 0, y > 0)\)

**Proof:**Let \(\log_b x = m \Rightarrow b^m = x\) and \(\log_b y = n \Rightarrow b^n = y\)\(x.y = b^m.b^n = b^{m + n} = b^{o}(say)\)

\(m + n = o\)

\(\log_b{x.y} = \log_b x + \log_b y\)

**Corollary:**\(\log_m(x.y.z) = \log_b x + \log_b y + \log_b z\)If \(x <0, y< 0, \log_b (x.y) = \log_b |x| + \log_b |y|\)

\(\log_b\left(\frac{x}{y}\right) = \log_b x - log_b y [x > 0, y > 0]\)

**Proof:**Let \(\log_b x = m \Rightarrow b^m = x\) and \(\log_b y = n \Rightarrow b^n = y\)\(\log_b\left(\frac{x}{y}\right) = o\Rightarrow b^o = \frac{x}{y}\)

\(\frac{x}{y} = \frac{b^m}{b^n} = b^{m - n} = b^o \Rightarrow m - n = o\)

\(\log_b\left(\frac{x}{y}\right) = \log_b x - log_b y\)

\(\log_b\left(\frac{x}{y}\right) = \log_b |x| - log_b |y|~[x < 0, y < 0]\)

\(\log_b N^k = k.\log_b N\)

**Proof:**Let \(\log_b N = x \Rightarrow b^x = N\)Let \(\log_b N^k = y \Rightarrow b^y = N^k \Rightarrow b^y = b^{x^k} = b^{kx}\)

\(y = kx \Rightarrow \log_b N^k = k.\log_b N\)

\(\log_b a = \log_c a\log_b c\)

**Proof:**Let \(\log_b a = x~\therefore b^x = a\)\(\log_c a = y \therefore c^y = a\)

\(\log_b c = z \therefore b^z = c\)

\(b^x = a = c^y = b^{yz} \Rightarrow x = yz[\because b \neq 1]\)

Alternatively we can also write it as \(\log_b a = \frac{\log_c a}{\log_c b}\)

\(\log_{(b^k)} N = \frac{1}{k}\log_b N [b > 0]\)

**Proof:**From previous point we can infer that \(\log_{(b^k)} N = \frac{\log N}{\log b^k} = \frac{\log N}{k.\log b} = \frac{1}{k}\log_b N\)\(\log_{(b^k)} N = \frac{1}{k}\log_|b| N [b < 0, k = 2m, m\in N]\)

\(\log_b a = \frac{1}{\log_a b}\)

**Proof:**Let \(\log_b a = x \therefore b^x = a\)\(\log_a b = y \therefore a^y = b\)

\(a = b^y = a^{xy} \Rightarrow xy = 1\)

\(\Rightarrow \log_b a \log_a b = 1\)

## 9.3. Bases of Logarthims

There are two popular bases for logarithms. Common base is \(10\) and
another is \(e\). When base is \(10,\) logarithm is known as *common
logarithm* and when base is \(e,\) logarithm is known as *natural* or
*Napierian* logarithm.

\(\log_10 x\) is also written as \(lg~x\) and \(\log_e x\) as \(ln~x\)

## 9.4. Characteristics and Mantissa

Typically a logarithm will have an integral part and a fractional part. The
integral part is called *characteristics* and fractional part is called
*mantissa.*

For example, if \(\log x = 4.7\) then \(4\) is characteristics and \(.7\) is mantissa of algorithm. If characteristics is less that zero then at times it is written with a bar above it. For example, \(\log x = -5.3 = \overline{5}.3\)

As you can easily figure out the number of possitive integers having base \(b\) and characteristics \(n\) is \(b^{n + 1} - b^n.\)

## 9.5. Inequality of Logarithms

If \(b > 1,\) and \(\log_b x_1 > \log_b x_2\) then \(x_1 > x_2.\) If \(b < 1,\) and \(\log_b x_1 > \log_b x_2\) then \(x_1 < x_2.\)

## 9.6. Expansion of Logarithm and its Graph

The logarithm series is given below:

\(\log(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots\)

Given below is an example how logarithm function behaves.

So you can see the rate of increment of logarithm decreases.