unreasonably effective

you can be sloppy, as long as you are rigorous

logarithmic function

Logarithmic function is the inverse of exponential function. Duh! Yet, this all important definition I have stashed too far away from my conscious. By practice it has become second nature to write: \log_b xy = \log_b x + \log_b y, which also helps explain \log_b a^n = n \log_b a or \log_b \frac{x}{y} = \log_b x - \log_b y. When in doubt I might go back to the fact: \log_bx = y \Rightarrow b^y = x. But I would have trouble explaining why logarithmic function is not defined for 0 or negative real numbers or where does the product-to-sum formula comes from or where my ultimate savior \log_bx = y \Rightarrow b^y = x comes from. They all come from logarithmic function being the inverse of exponential function.

Given we know about exponential function and the general properties of inverse function, we may derive all that is required. Exponential function is like repeated multiplication and thus easier to think about. Although inverse function of a one-to-one function might be confusing but the payoff is great, once learned, we could derive properties of inverse functions easily and thus greatly extend our function repertoire.

For example the complicated identity: a^{\log_a x} = x is actually a statement of the simple fact: f^{-1}\left(f(x)\right) = x where f(x) = \log_a x and f^{-1}(x) = a^x.

An inverse function has domain \equiv original function’s range and inverse function’s range \equiv original function’s domain. Since the exponential function f(x) = a^x with a > 0 cannot be zero or negative for any real number x, it’s range is (0, \infty) which says logarithm function–the inverse of exponential–has domain (0, \infty). Therefore, a logarithm function for base a is not defined for 0 or negative numbers.

Passing both sides of \log_b x = y into the exponential function f(x) = b^x we readily get b^{\log_b x} = b^y. Since exponential function, being the inverse of logarithmic function, cancels logarithm’s effect, we get: x = b^y.

We may remember the product-to-sum also happens in exponential function: a^{x} \cdot a^{y} = a^{x+y}. With the goal of using this fact, in \log_b xy, we may express x as b^{\log_b x} and we may express y as b^{\log_b y}. Then \log_b xy = \log_b \left( b^{\log_b x} \cdot b^{\log_b y} \right) = \log_b \left( b^{\left(\log_b x+ \log_b y\right)} \right) = \log_b x + \log_b y.

Logarithm was invented for it’s product-to-sum property which turns multiplication into addition and thus helps in manual computation. However, now-a-days, that is not the reason we still learn it. It pops up in unexpected places. For example, log-normal distribution, entropy of information, distribution of prime numbers involve logarithmic function as does the infinite sum \sum_{n=1}^{\infty} \frac{1}{n}.

tanvirdotzaman

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