Expanding & Condensing Logarithms Worksheet Answers Simplified
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Expanding and condensing logarithms can often be a complex topic for students grappling with algebra. Understanding these concepts is crucial for solving logarithmic equations effectively. In this article, we’ll simplify the concept of expanding and condensing logarithms and provide practical examples along with solutions.
What are Logarithms?
Logarithms are the inverse operations of exponentiation. A logarithm answers the question: "To what exponent must a base be raised to obtain a certain number?" For example, in the expression ( \log_b(a) = c ), the base ( b ) raised to the power ( c ) equals ( a ).
The Properties of Logarithms
Understanding the properties of logarithms is vital for both expanding and condensing logarithmic expressions. Here are the main properties to keep in mind:
- Product Rule: [ \log_b(M \cdot N) = \log_b(M) + \log_b(N) ]
- Quotient Rule: [ \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) ]
- Power Rule: [ \log_b(M^p) = p \cdot \log_b(M) ]
These properties help in manipulating logarithmic expressions to either expand or condense them.
Expanding Logarithms
Expanding a logarithm means expressing it as a sum or difference of simpler logarithmic terms. Let’s look at some examples.
Example 1: Expanding Using the Product Rule
Given: [ \log_b(5x) ]
Solution: Using the product rule, we can expand this expression: [ \log_b(5) + \log_b(x) ]
Example 2: Expanding Using the Quotient Rule
Given: [ \log_b\left(\frac{10}{y}\right) ]
Solution: Applying the quotient rule, we get: [ \log_b(10) - \log_b(y) ]
Example 3: Expanding Using the Power Rule
Given: [ \log_b(3^4) ]
Solution: Using the power rule: [ 4 \cdot \log_b(3) ]
Summary of Expanding Logarithms
<table> <tr> <th>Logarithmic Expression</th> <th>Expanded Form</th> </tr> <tr> <td>( \log_b(5x) )</td> <td> ( \log_b(5) + \log_b(x) ) </td> </tr> <tr> <td>( \log_b\left(\frac{10}{y}\right) )</td> <td> ( \log_b(10) - \log_b(y) ) </td> </tr> <tr> <td>( \log_b(3^4) )</td> <td> ( 4 \cdot \log_b(3) ) </td> </tr> </table>
Condensing Logarithms
Condensing logarithms is the reverse process. It involves taking the sum or difference of logarithmic terms and expressing them as a single logarithm.
Example 1: Condensing Using the Product Rule
Given: [ \log_b(5) + \log_b(x) ]
Solution: By applying the product rule in reverse: [ \log_b(5x) ]
Example 2: Condensing Using the Quotient Rule
Given: [ \log_b(10) - \log_b(y) ]
Solution: Using the quotient rule in reverse: [ \log_b\left(\frac{10}{y}\right) ]
Example 3: Condensing Using the Power Rule
Given: [ 4 \cdot \log_b(3) ]
Solution: Using the power rule in reverse: [ \log_b(3^4) ]
Summary of Condensing Logarithms
<table> <tr> <th>Logarithmic Expression</th> <th>Condensed Form</th> </tr> <tr> <td>( \log_b(5) + \log_b(x) )</td> <td> ( \log_b(5x) ) </td> </tr> <tr> <td>( \log_b(10) - \log_b(y) )</td> <td> ( \log_b\left(\frac{10}{y}\right) ) </td> </tr> <tr> <td>( 4 \cdot \log_b(3) )</td> <td> ( \log_b(3^4) ) </td> </tr> </table>
Important Notes to Remember
"Always remember to apply the properties of logarithms correctly while expanding or condensing. Mistakes in these steps can lead to incorrect conclusions."
Practice Problems
To master expanding and condensing logarithms, practice is essential. Here are some problems for you to try:
- Expand: ( \log_b(2xy) )
- Condense: ( \log_b(6) + \log_b(4) - \log_b(2) )
- Expand: ( \log_b\left(\frac{x^3y}{z^2}\right) )
- Condense: ( 2\log_b(5) + \log_b(3) )
Answers
- ( \log_b(2) + \log_b(x) + \log_b(y) )
- ( \log_b(12) )
- ( \log_b(x^3) + \log_b(y) - \log_b(z^2) )
- ( \log_b(25) + \log_b(3) = \log_b(75) )
Understanding and practicing the properties of logarithms will significantly enhance your mathematical skills, helping you tackle more complex problems in algebra and beyond. Happy studying! 😊