Logarithmic Functions

**Logarithmic Functions**

**Logarithmic functions**, or logs, are the inverse of exponential functions. The two can be written in either form.

*y* = *b** ^{x}* can be rewritten as log

*(*

_{b}*y*) =

*x*

The main thing to remember about log functions is that they produce exponents.

**EXAMPLES**

Evaluate log_{2} 32.

The question here is basically, "What exponent of 2 will give you 32?" Remembering that 2^{5} = 32, it follows that log_{2} 32 = **5**.

Evaluate log_{3} 9.

Don't be fooled here. This is not a division problem; the question is asking for an *exponent*. Since 3^{2} = 9, this means log_{3} 9 = **2**.

Evaluate log_{5} (^{1}/_{25}).

Remembering that negative exponents make reciprocals, we find that 5^{-2} = ^{1}/_{25}. So log_{5} (^{1}/_{25}) = **-2**.

Evaluate log_{7} (-49).

There is no such exponent that satisfies 7^{?} = -49. Therefore, log_{7} (-49) **does not exist**.

Using the definition of logarithm, you can also solve equations.

EXAMPLE

Solve: log_{2}(*x*) = 4

Understand:

This reads "the log base two of a number equals four." Solve by converting the logarithmic statement into its equivalent exponential form.

Solution:

log_{2}(*x*) = 4

2^{4} = *x*

**16 = x**

You can also get the graph of a log function from the graph of an exponential function.

Recall how *y* = 2* ^{x}* looks on the graph:

The domain is all real numbers, the range is only the *positive* numbers, there's a *y*-intercept at 1, and there's a horizontal asymptote through *y* = 0.

If we think about reflecting this graph over the diagonal line *y* = *x*, we get the graph of *y* = log_{2}(*x*).

Notice with this function, the domain is only the positive numbers (remember how log_{7} (-49) didn't exist?), the range is all real numbers, there's an *x*-intercept at 1, and there's a *vertical* asymptote through *x* = 0.

Like with exponential functions, the appearance of the log function graph varies depending on the base, but all log graphs have the same general shape:

*y* = log_{2} (*x*)

*y* = log_{3} (*x*)

*y* = log_{5} (*x*)

*y* = log_{10} (*x*)

The base 10 logarithm is also known as the *common logarithm*. If the base isn't written, it's understood to be 10.

log (*x*) = log_{10} (*x*)