Hi everyone!! Its me David.. I'm pretty sure some of you already knew I have an identical twin brother named Daniel, so I want the rest of you to know that I don't have a doppelganger out there.. (dop·pel·gäng·er : someone who looks like someone else)

Logarithmic Functions

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

loga x = y means a y = x .

where a is called the base; a > 0, x > 0 and a≠1 . For example, log232 = 5 because25 = 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 = log 7343 . Then y = 3 .
y = log 10100000 . Then y = 5 .
y = log 10164 . Then using a log table or calculator, y  2.215 .
y = log 4276 . 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) = log2 x looks like:

 f (x) = log2 x

The graph of f (x) = log2 x has a vertical asymptote at x = 0 and passes through the point (1, 0) .

We can see that f(x) = log2 x is the inverse of g(x) = 2x because f (x) is the reflection of g(x) over the line y = x :

 f (x) = log2 x and g(x) = 2x

f (x) = loga x 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.