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Logarithmic Functions

Like many types of functions, the exponential function has an inverse. This inverse is called the logarithmic function.

logax = y means ay = x.
where a is called the base; a > 0 and a≠1. For example, log232 = 5 because 25 = 32. log5 = - 3 because 5-3 = .

To evaluate a logarithmic function, determine what exponent the base must be taken to in order to yield the number x. Sometimes the exponent will not be a whole number. If this is the case, consult a logarithm table or use a calculator.


Examples:
y = log39. Then y = 2.
y = log5. Then y = - 4.
y = log. Then y = 3.
y = log7343. Then y = 3.
y = log10100000. Then y = 5.
y = log10164. Then using a log table or calculator, y 2.215.
y = log4276. Then using a log table or calculator, y 4.054.

Since no positive base to any power is equal to a negative number, we cannot take the log of a negative number.

The graph of f (x) = log2x looks like:

Figure %: f (x) = log2x
The graph of f (x) = log2x has a vertical asymptote at x = 0 and passes through the point (1, 0).

Note that f (x) = log2x is the inverse of g(x) = 2x. fog(x) = log22x = x and gof (x) = 2log2x = x (we will learn why this is true in Log properties). We can also see that f (x) = log2x is the inverse of g(x) = 2x because f (x) is the reflection of g(x) over the line y = x:

Figure %: f (x) = log2x and g(x) = 2x
f (x) = logax can be translated, stretched, shrunk, and reflected using the principles in Translations, Stretches, and Reflections.

In general, f (x) = c·loga(x - h) + k has a vertical asymptote at x = h and passes through the point (h + 1, k). The domain of f (x) is and the range of f (x) is . Note that this domain and range are the opposite of the domain and range of g(x) = c·ax-h + k given in Exponential Functions.

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