Given that [tex]$\log _a x=3, \log _a y=4$[/tex], and [tex]$\log _a z=5$[/tex], find the following.
[tex]$\log _a \frac{\sqrt[3]{y^2 z^4}}{\sqrt[3]{x^2 z^{-4}}}$[/tex]
Answer (1)
Use logarithm properties to expand the expression: lo g a 3 x 2 z − 4 3 y 2 z 4 = 3 1 ( 2 lo g a y + 4 lo g a z ) − 3 1 ( 2 lo g a x − 4 lo g a z ) .
Substitute the given values: lo g a x = 3 , lo g a y = 4 , and lo g a z = 5 .
Simplify the expression: 3 1 ( 2 ( 4 ) + 4 ( 5 )) − 3 1 ( 2 ( 3 ) − 4 ( 5 )) = 3 1 ( 28 ) − 3 1 ( − 14 ) .
Calculate the final result: 3 28 + 3 14 = 3 42 = 14 . The answer is 14 .
Explanation
Understanding the Problem We are given the values of lo g a x , lo g a y , and lo g a z , and we want to find the value of the expression lo g a 3 x 2 z − 4 3 y 2 z 4 . We will use properties of logarithms to simplify the expression and then substitute the given values.
Simplifying the Expression First, we can rewrite the expression using the properties of logarithms: lo g a 3 x 2 z − 4 3 y 2 z 4 = lo g a ( y 2 z 4 ) 1/3 − lo g a ( x 2 z − 4 ) 1/3 = 3 1 lo g a ( y 2 z 4 ) − 3 1 lo g a ( x 2 z − 4 ) = 3 1 ( lo g a y 2 + lo g a z 4 ) − 3 1 ( lo g a x 2 + lo g a z − 4 ) = 3 1 ( 2 lo g a y + 4 lo g a z ) − 3 1 ( 2 lo g a x − 4 lo g a z )
Substituting the Values and Calculating Now, we substitute the given values lo g a x = 3 , lo g a y = 4 , and lo g a z = 5 into the expression: = 3 1 ( 2 ( 4 ) + 4 ( 5 )) − 3 1 ( 2 ( 3 ) − 4 ( 5 )) = 3 1 ( 8 + 20 ) − 3 1 ( 6 − 20 ) = 3 1 ( 28 ) − 3 1 ( − 14 ) = 3 28 + 3 14 = 3 42 = 14
Final Answer Therefore, lo g a 3 x 2 z − 4 3 y 2 z 4 = 14 .
Examples
Logarithms are used in many scientific fields, such as physics, chemistry, and engineering. For example, the Richter scale, which measures the magnitude of earthquakes, is a logarithmic scale. This means that an earthquake of magnitude 6 is ten times stronger than an earthquake of magnitude 5. Logarithms are also used in computer science to analyze the complexity of algorithms. Understanding and manipulating logarithmic expressions is therefore a valuable skill in many areas.
