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We have seen that, with a base 0 < a ¹ 1, the exponential function f(x) = ax is defined on R having range (0, ∞). We also observed that f(x) is a bijection, hence it has an inverse. We call this inverse function as logarithmic function and is denoted by loga(.). Let us discuss this function further. Note that if f(x) takes x to y = ax, then loga(.) takes y to x. That is, for 0 < a ¹ 1, we have
y = ax is equivalent to loga y = x.
For example, since 34 = 81 we have log3(81) = 4. In other words, with fixed a, given a real number y, logarithm finds the exponent x satisfying ax = y. This is useful in addressing practical problems like, “how long will it take for certain investment to reach a fixed amount?” Logarithm is also very useful in multiplying very small or big numbers.
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