Difference between revisions of "Logarithms"
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− | + | ===Logarithms=== | |
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− | + | ===Graphing a Logarithmic Function=== | |
+ | [[Image:Log.png|thumb|200px|Graph of a Logarithmic Function]] | ||
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+ | This is a graph of a basic logarithmic function: y = log(x). A logarithmic function, such as the one used to create the featured image, takes the basic form <math>y = log_b(x)</math>, where ''b'' is fixed while ''y'' and ''x'' are variables. | ||
+ | =Basic Description= | ||
− | + | Logarithms are considered the inverse or opposite operations to exponents, just as subtraction is the inverse to to addition or square rooting is the inverse to squaring. | |
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− | We could do the inverse operation by using logarithms and write <math> log_{2}(8) = 3</math> , which is read "logarithm of 8 base 2 is equal to 3". | + | For example, suppose we have the exponential expression <math>2^3</math> , which we know will equal 8. Now, suppose we want to do the inverse operation and go from the value 8 to the exponent 3 with a base of 2. We could do the inverse operation by using logarithms and write <math> log_{2}(8) = 3</math> , which is read "logarithm of 8 base 2 is equal to 3". |
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− | What about exponential equations in the form <math>10^x = 932</math>? | + | What about exponential equations in the form <math>10^x = 932</math>? It might seem harder to solve for x in this case because there is no <balloon title="load:exp">whole number exponent</balloon><span id="exp" style="display:none"><math>10^2 = 100...... \,</math><math>10^x = 932...... \,</math> <math>10^3 = 1000\,</math></span> that will give us the value of 932 with a base of 10. However, if we simply rewrite the equation as an logarithmic equation <math>x = log_{10}(932)</math> , we can find quite easily with a calculator that x is about 2.969. |
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− | It might seem harder to solve for x in this case because there is no <balloon title="load:exp">whole number exponent</balloon><span id="exp" style="display:none"><math>10^2 = 100...... \,</math><math>10^x = 932...... \,</math> <math>10^3 = 1000\,</math></span> that will give us the value of 932 with a base of 10. | ||
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− | However, if we simply rewrite the equation as an logarithmic equation <math>x = log_{10}(932)</math> , we can find quite easily with a calculator that x is about 2.969. | ||
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+ | =A More Mathematical Description= | ||
==Definition of Logarithms== | ==Definition of Logarithms== | ||
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<tr><td> <math>log_a(b) = \frac{1}{log_b(a)}\,</math></td></tr> | <tr><td> <math>log_a(b) = \frac{1}{log_b(a)}\,</math></td></tr> | ||
</table> | </table> | ||
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Revision as of 09:55, 10 June 2009
Contents
Logarithms
Graphing a Logarithmic Function
This is a graph of a basic logarithmic function: y = log(x). A logarithmic function, such as the one used to create the featured image, takes the basic form , where b is fixed while y and x are variables.
Basic Description
Logarithms are considered the inverse or opposite operations to exponents, just as subtraction is the inverse to to addition or square rooting is the inverse to squaring.
For example, suppose we have the exponential expression , which we know will equal 8. Now, suppose we want to do the inverse operation and go from the value 8 to the exponent 3 with a base of 2. We could do the inverse operation by using logarithms and write , which is read "logarithm of 8 base 2 is equal to 3".
- In order words, if we have an exponential equation:
- we can write an equivalent logarithmic equation:
What about exponential equations in the form ? It might seem harder to solve for x in this case because there is no whole number exponent that will give us the value of 932 with a base of 10. However, if we simply rewrite the equation as an logarithmic equation , we can find quite easily with a calculator that x is about 2.969.
To look at some more examples of switching between exponential and logarithmic equations:
A More Mathematical Description
Definition of Logarithms
if and only if , where b > 1 and x > 0
In words: The logarithm of a value at a given base is the power (exponent) that the base must be raised to produce the value.
As you can see from the graph, there is an vertical asymptote at x = 0, so that logarithmic functions are undefined when x is less than or equal to 0. However, non-real logarithms for negative x values can be found using complex logarithms with complex numbers.
Bases
As seen from the definition above, the base of a logarithm affects how a logarithm is evaluated. Bases can be any positive number except for 1, and the logarithms of a value can be found at different bases using a change of base formula.
Common Bases
There are three main bases that are most frequently used:
Base | Exponential Representation | Logarithmic Representation | Notes | Example |
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Base 10 | can be written simply as
also called Common Logarithms |
where x = 2 |
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Base 2 | basis for the Binary System |
where x = 4 |
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Base e | can be written simply as
also called Natural Logarithms where |
where |
Changing Bases
To go from a logarithm of base k to a logarithm of base a, we use the formula:
Basic Properties of Logarithms
Logarithms possess various properties and identities including the following:
Identities | |
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Multiplication | |
Division | |
Exponentiation | |
Integration | |
Differentiation |
Other properties |
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