Logarithmic Functions

Given the exponential function

f(x) = y = b^x

it is often desirable to solve for x in terms of y. In order to accomplish this, we define a new function called the logarithm (log for short) to the base b, b > 0 , defined as follows:

g(x) = y = log _b x whenever x = b^y

That is, the logarithm to the base b of a real number x is the real number y such that b raised to the yth power is equal to x. The quantity b^y, where b > 0 , will always have a positive value for any real number y.

If x = b^y, then y = log _b x and x must be positive in either equality. Thus, the logarithm function has a domain that consists only of positive real numbers.

Note that if b = 1, then b^y = 1 for any real number y. Thus, the logarithm function having base 1 has only one element in its domain, the number 1, and so it has no real significance.

The quantity log _b x, where b = 10, is often referred to as simply log x.

The functions f(x) = b^x and g(x) = log b x are inverse functions of each other. To verify this, note that for f(x) = b^x and g(x) = log _b x ,

f(g(x)) = b^{log3 x}
z = log _b x implies that b^z = x

b^{log3 x} = x

g(f(x)) = log _bb^x = x since b^x = b^x.

EX. log ₅ 25 = 2 since 5² = 25
log ₁₀ 0.00001 = -5 since 10^-5 = 0.00001
log _0.5 0.0625 = 4 since (0.5)⁴ = 0.0625

Since the functions f(x) = b^x and g(x) = log _b x are inverse functions of each other, their graphs are mirror images of each other across the line y = x. The graph of g(x) = log _b x, where b > 0 and b ¹ 1, will also curve to the right and move upwards without bound as x goes to infinity, although more slowly than the graph f(x) = b^x. It will approach the y-axis asymptotically as x approaches 0, since b^x® 0 as x ® -¥ and thus

log _b x ® -¥ as x ® 0

EX. The logarithmic function

f(x) = log ₃ x

contains the following points: