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.

Figure made with TikZ

Plot of logarithm with base 2

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