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:

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

2. 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 $$b \neq 0$$

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

4. 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.

5. 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 $$b^y = N$$

$$\Rightarrow b^x = b^y \Rightarrow x = y$$

6. 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

1. $$\log_b 1 = 0$$

Proof: Let $$\log_b 1 = x\Rightarrow b^x = 1 \Rightarrow x = 0$$

2. $$=log_b b = 1$$

Proof: Let $$\log_b b = x\Rightarrow b^x = b\Rightarrow x = 1$$

3. $$b^{\log_b N} = N$$

Proof: Let $$\log_b N = x \Rightarrow b^x = N$$

$$b^{\log_b N} = N [\because x = \log_b N]$$

9.2. Important Formulae

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

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 $$x <0, y< 0, \log_b (x.y) = \log_b |x| + \log_b |y|$$

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

Proof: Let $$\log_b x = m \Rightarrow b^m = x$$ and $$\log_b y = n \Rightarrow b^n = y$$

$$\log_b\left(\frac{x}{y}\right) = o\Rightarrow b^o = \frac{x}{y}$$

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

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

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

3. $$\log_b N^k = k.\log_b N$$

Proof: Let $$\log_b N = x \Rightarrow b^x = N$$

Let $$\log_b N^k = y \Rightarrow b^y = N^k \Rightarrow b^y = b^{x^k} = b^{kx}$$

$$y = kx \Rightarrow \log_b N^k = k.\log_b N$$

4. $$\log_b a = \log_c a\log_b c$$

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 $$\log_b a = \frac{\log_c a}{\log_c b}$$

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

Proof: From previous point we can infer that $$\log_{(b^k)} N = \frac{\log N}{\log b^k} = \frac{\log N}{k.\log b} = \frac{1}{k}\log_b N$$

$$\log_{(b^k)} N = \frac{1}{k}\log_|b| N [b < 0, k = 2m, m\in N]$$

6. $$\log_b a = \frac{1}{\log_a b}$$

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.