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Logarithmic Functions

Logarithmic Functions

Given the exponential function

f(x) = y = bx

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 = by

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 by, where b > 0 , will always have a positive value for any real number y.

If x = by, 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 by = 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) = bx and g(x) = log b x are inverse functions of each other. To verify this, note that for f(x) = bx and g(x) = log b x ,

f(g(x)) = blog3 x
z = log b x implies that bz = x

blog3 x = x

g(f(x)) = log bbx = x since bx = bx.

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) = bx 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) = bx. It will approach the y-axis asymptotically as x approaches 0, since bx® 0 as x ® -¥ and thus

log b x ® -¥ as x ® 0

EX. The logarithmic function

f(x) = log 3 x

contains the following points:

x log 3 x (x,y)
0.004 -5 (0.004, -5)
0.012 -4 (0.012, -4)
0.037 -3 (0.037, -3)
0.111 -2 (0.111, -2)
0.333 -1 (0.333, -1)
1 0 (1, 0)
3 1 (3, 1)
9 2 (9, 2)
27 3 (27, 3)
81 4 (81, 4)
243 5 (243, 5)
Subject: 
Subject X2: 

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