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:

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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:

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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:

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4. Change of Base Rule:

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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. Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

math.brown.edu, June 18, 2008, Created with GeoGebra

SOLVING LOG EQUATIONS: (top)

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PRACTICE PROBLEMS: (answers | solutions | top)

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