## Logarithms

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Logarithms: Basics :: Log Rules :: Graphing :: Practice Problems :: Quiz

**BASICS:** (top)

A **logarithm** is the power to which a base must be raised to obtain a certain result. The exponential form can be rewritten with the logarithmic form . This is read as "y equals log base b of x."

EXAMPLE 1:

Problem:Solve the logarithmic equation

Solution:This logarithmic equation is equivalent to the exponential equation . Since , . Thus, . Saying "log base 3 of 9 equals 2" means that raising 3 to the 2nd power equals 9.

Note that log base 10 is known as the **common log**; if log is written without a base, it usually implies log base 10. Likewise, log base e is written using the notation ln, read as "**natural log**." (Often in higher mathematics, log without a base also refers to log base e).

Note also that . We know means the power to which b is raised to get x, so raising b to the power to which b is raised to get x gives x.

**LOGARITHM RULES:** (top)

It can be very useful to rewrite logarithmic functions in different ways. This section outlines the rules that make such rearragements possible.

1. Product Rule:

EXAMPLE 1:

Problem:Solve the logarithmic equation .

Solution:Note that both logarithms have the same base, so they can be combined using the multiplication rule to give . Now that there's only one logarithm, we can change it to the exponential form to get . All that's left to do is simplify: .

2. Quotient Rule:

EXAMPLE 2:

Problem:Solve the logarithmic equation .

Solution:Again, both logarithms have the same base so they can be combined, this times using the quotient rule to give . Now that there's only one logarithm, we can change it to the exponential form to get . All that's left to do is simplify: .

3. Power Rule:

4. Change of Base Rule:

EXAMPLE 3:

Problem:Express as a natural log.

Solution:We use the change of base rule to convert the logarithm from base 3 to base e. We get .

**GRAPHING:** (top)

The graphs of logarithmic functions are essentially sideways versions of exponential decay functions. The graph of f(x)=ln(x) is shown below.

Note that the domain here is x>0; you cannot take a log of 0 or a negative number. This makes sense if you consider what taking a logarithm of, say, x means: it is the power to which the base must be raised to get x. There is no exponential power that we can raise a base to to get 0. Likewise, there is no exponential power that we can raise a base to to get any negative number; recall that the range of a standard exponential function was y>0.

The range of a logarithmic function is all real numbers. Though the function increases more and more slowly as x gets bigger, it does keep increasing. Thus, it is also the case

The demo below shows the effects of changing different aspects of the logarithmic equation y=aln(bx+c)+d. Use the sliders to change variables a, b, c, and d.
math.brown.edu, June 18, 2008, Created with GeoGebra |

**SOLVING LOG EQUATIONS:** (top)

**PRACTICE PROBLEMS:** (answers | solutions | top)

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This page last updated 18 June, 2008 2:18 PM