Identify the properties used to perform the following simplifications:
[tex]\begin{array}{l}
\log (100)=\log (400)-\log (4) \\
\log (x^2)=\log (x)+\log (x) \\
\log (49)=2 \\log (7)
\end{array}[/tex]
Which properties would you use to simplify the following expression?
[tex]\log \left(17 x^3\right)[/tex]
Properties:
1. [tex]\log (x y)=\log x+\log y[/tex]
2. [tex]\log \left(\frac{x}{y}\right)=\log x-\log y[/tex]
3. [tex]\log \left(x^r\right)=r \\log x[/tex]
Answer (1)
The expression lo g ( 17 x 3 ) can be simplified using the following properties:
Apply the product rule: lo g ( x y ) = lo g x + lo g y , which gives lo g ( 17 x 3 ) = lo g ( 17 ) + lo g ( x 3 ) .
Apply the power rule: lo g ( x r ) = r lo g x , which gives lo g ( x 3 ) = 3 lo g ( x ) .
Combine the results: lo g ( 17 x 3 ) = lo g ( 17 ) + 3 lo g ( x ) .
Thus, properties 1 and 3 are used to simplify the expression. 1 , 3
Explanation
Analyzing the Expression and Properties We are given the expression lo g ( 17 x 3 ) and asked to identify which logarithmic properties can be used to simplify it. We have three properties to consider:
lo g ( x y ) = lo g x + lo g y
lo g ( y x ) = lo g x − lo g y
lo g ( x r ) = r lo g x
Applying the Product Rule First, we can apply property 1, the product rule, to the expression lo g ( 17 x 3 ) . This allows us to separate the product of 17 and x 3 :
lo g ( 17 x 3 ) = lo g ( 17 ) + lo g ( x 3 ) So, property 1 is applicable.
Applying the Power Rule Next, we look at the term lo g ( x 3 ) . We can apply property 3, the power rule, to bring the exponent 3 down as a coefficient: lo g ( x 3 ) = 3 lo g ( x ) So, property 3 is also applicable.
Final Simplification and Conclusion Combining these results, we have: lo g ( 17 x 3 ) = lo g ( 17 ) + lo g ( x 3 ) = lo g ( 17 ) + 3 lo g ( x ) Therefore, properties 1 and 3 are used to simplify the expression. Property 2, the quotient rule, is not applicable here because we do not have a quotient within the logarithm.
Examples
Logarithmic properties are useful in many fields, including physics and engineering. For example, the Richter scale, which measures the magnitude of earthquakes, is a logarithmic scale. An earthquake of magnitude 6 is ten times stronger than an earthquake of magnitude 5. Similarly, in acoustics, the decibel scale uses logarithms to measure sound intensity. Understanding logarithmic properties helps scientists and engineers work with these scales and analyze data effectively.
