Exponential and Logarithmic Functions

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If f is a nonconstant function that is continuous and satisfies the functional equation f(x+y) = f(x) * f(y), then f(x) = a^x for some constant a. Note that * denotes multiplication.

Consider the exponential function a^x, a > 0 and the logarithmic function log_a x, a > 0. Then a^x is defined for all x in R, and log_a x is defined only for positive x in R.

Now, let a^x, a > 0 be an exponential function. Then for any real numbers x and y
a^x * a^y = a^x+y
(a^x)^y = a^xy
Next, let lob_a x, a > 0 be a logarithmic function. Then for any positive real numbers x and y
log_a(xy) = log_a(x) + log_a(y)
log_a(x^y) = y log_a(x)