8d)++Properties+of+Logarithms

In this lesson you will learn the different properties of Logarithms. Properties of logarithms allow us to explore how logarithms work and allows us to expand and condense logarithm expressions. The first basic property of logarithms is log b x = y, where "b" is the base, "y" is the exponent that "b" is raised to, and "x" is the solution to "b" raised to the "y" power. Now that you know the basic property, we can expand logarithms in this form. There is three identity properties: The first identity shows how to __expand__ a logarithm when you have two logs that are multiplied together. In this equation, you take the log of the first term and add it to the log of the second term. The second identity shown is how to expand logs with division. In this case, you take the log of the numerator and subtract it from the log of the denominator. The last Identity shown demonstrates how to expand logarithm expressions with an exponent. When an exponent is present, you take the log of x, and take the exponent and move it in front of the entire log.
 * Product property:**
 * Division Property:**
 * Exponent Property:**

To __condense__ a logarithm, you do the steps in revers, so to condense the first example, for the first example, you take log of x and the log of y and multiply the two together to get the log(xy), for the second example you take the log of x divided by y and for the last example, you take the number in front of the entire log and place it in the exponent position for the term x.

Bellow are two links further explaining the different properties of logarithms with examples and practice activities. Under the links are more examples on how each property works, especially when more than one property must be used to expand or condense an equation.

[|site 1] [|site 2]

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CLICK HERE FOR THE TUTORIAL!!
Condensing and Expanding Using Product Property: **  **  Condensing and Expanding Using Division Property:  Expand and solve: To expand, first separate the fraction. Writing out Log and the base for each, the fraction can be divided by using subtraction. To solve, use the change of base formula for the two logs individually Subtract.
 * EXAMPLES:

Condensing: the logs have the same base, so you can put the 9 over the 3 Simplify!

To condense, logs must have the same base. Once the bases are the same, if subtraction is present, than you can create a fraction. This is because of the quotient property

Here try one for yourself: 1) **

2)

Answers: 1) you get the decimal by using the change of base formula with your calculator 2) Simply set it as a fraction because they have the same base, than simplify Use the change of base formula (as shown) and get your answer!
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Condensing and Expanding Using Exponent Property:



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