Expand $\log _7\left(x^3 / y\right)^3$
A. $9 \log _7 x-3 \log _7 y$
B. $3 \log _7 x-3 \log _7 y$
C. $9 \log _7 x-\log _7 y$
D. $9 \log _7 x+3 \log _7 y$.
Answer (1)
Apply the power rule: lo g 7 ( x 3 / y ) 3 = 3 lo g 7 ( x 3 / y ) .
Apply the quotient rule: 3 lo g 7 ( x 3 / y ) = 3 ( lo g 7 ( x 3 ) − lo g 7 ( y )) .
Apply the power rule again: 3 ( lo g 7 ( x 3 ) − lo g 7 ( y )) = 3 ( 3 lo g 7 ( x ) − lo g 7 ( y )) .
Distribute: 3 ( 3 lo g 7 ( x ) − lo g 7 ( y )) = 9 lo g 7 x − 3 lo g 7 y . The final answer is 9 lo g 7 x − 3 lo g 7 y .
Explanation
Understanding the Problem We are asked to expand the logarithmic expression lo g 7 ( x 3 / y ) 3 . We need to use properties of logarithms to expand the expression. The properties of logarithms that will be used are: lo g b ( a c ) = c lo g b ( a ) and lo g b ( a / c ) = lo g b ( a ) − lo g b ( c ) .
Applying the Power Rule We will use the power rule of logarithms to rewrite the expression as 3 lo g 7 ( x 3 / y ) .
Applying the Quotient Rule Next, we use the quotient rule of logarithms to rewrite the expression as 3 ( lo g 7 ( x 3 ) − lo g 7 ( y )) .
Applying the Power Rule Again Then, we use the power rule of logarithms again to rewrite the expression as 3 ( 3 lo g 7 ( x ) − lo g 7 ( y )) .
Distributing and Final Answer Now, we distribute the 3 to get 9 lo g 7 ( x ) − 3 lo g 7 ( y ) . The final expanded form is 9 lo g 7 x − 3 lo g 7 y .
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (the Richter scale), the loudness of sounds (decibels), and the acidity of solutions (pH scale). Understanding how to expand and simplify logarithmic expressions is crucial in these contexts. For instance, in seismology, the magnitude of an earthquake is calculated using logarithms. If you have data about the amplitude of seismic waves and the distance to the epicenter, you can use logarithmic properties to determine the earthquake's magnitude. Similarly, in acoustics, the sound intensity level is measured in decibels, which is a logarithmic scale. Expanding logarithmic expressions helps in analyzing and comparing sound intensities from different sources. These applications demonstrate the practical significance of mastering logarithmic properties.
