Given that [tex]$\log _a x=3, \log _a y=4$[/tex], and [tex]$\log _a z=5$[/tex], find the following [tex]$\log _a \frac{\sqrt[4]{y^3 z^4}}{\sqrt[4]{x^4 z^{-4}}}$[/tex]
Answer (1)
Simplify the expression using properties of logarithms and exponents.
Substitute the given values of lo g a x , lo g a y , and lo g a z .
Calculate the final value.
The final answer is 10 .
Explanation
Understanding the Problem We are given that lo g a x = 3 , lo g a y = 4 , and lo g a z = 5 . We want to find the value of lo g a 4 x 4 z − 4 4 y 3 z 4 .
Simplifying the Expression First, let's simplify the expression using properties of exponents and radicals: lo g a 4 x 4 z − 4 4 y 3 z 4 = lo g a 4 x 4 z − 4 y 3 z 4 = lo g a ( x 4 z − 4 y 3 z 4 ) 4 1 Using the power rule of logarithms, we have: lo g a ( x 4 z − 4 y 3 z 4 ) 4 1 = 4 1 lo g a x 4 z − 4 y 3 z 4 Now, we use the quotient rule of logarithms: 4 1 lo g a x 4 z − 4 y 3 z 4 = 4 1 ( lo g a ( y 3 z 4 ) − lo g a ( x 4 z − 4 ) ) Using the product rule of logarithms, we have: 4 1 ( lo g a ( y 3 z 4 ) − lo g a ( x 4 z − 4 ) ) = 4 1 ( lo g a y 3 + lo g a z 4 − ( lo g a x 4 + lo g a z − 4 ) ) Using the power rule of logarithms again, we get: 4 1 ( 3 lo g a y + 4 lo g a z − ( 4 lo g a x + ( − 4 ) lo g a z ) ) = 4 1 ( 3 lo g a y + 4 lo g a z − 4 lo g a x + 4 lo g a z ) Combining like terms, we have: 4 1 ( 3 lo g a y + 8 lo g a z − 4 lo g a x )
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: 4 1 ( 3 ( 4 ) + 8 ( 5 ) − 4 ( 3 ) ) = 4 1 ( 12 + 40 − 12 ) = 4 1 ( 40 ) = 10 Therefore, \log _a \frac{\sqrt[4]{y^3 z^4}}{\sqrt[4]{x^4 z^{-4}}}} = 10 .
Final Answer Thus, the value of the expression is 10.
Examples
Logarithms are used extensively in various fields such as computer science, finance, and engineering. For example, in computer science, logarithms are used to analyze the time complexity of algorithms. In finance, they are used to calculate compound interest. In engineering, they are used in signal processing and control systems. Understanding how to simplify logarithmic expressions is crucial for solving problems in these fields. For instance, calculating the intensity of an earthquake using the Richter scale involves logarithmic calculations, where each whole number increase represents a tenfold increase in amplitude.
