What is $4 \log _{\frac{1}{2}} w+\left(2 \log _{\frac{1}{2}} u-3 \log _{\frac{1}{2}} v\right)$ written as a single logarithm?
A. $\log _{\frac{1}{2}} w^4 u^2-v^3$
B. $\log _{\frac{1}{2}} w^4\left(\frac{u^2}{v^3}\right)$
C. $\log _{\frac{1}{2}}\left(\frac{w^4}{u^2 v^3}\right)$
D. $\log _{\frac{1}{2}}\left(w\left(\frac{u^2}{v^3}\right)\right)^4$
Answer (1)
Use the power rule to rewrite the terms: 4 lo g 2 1 w = lo g 2 1 w 4 , 2 lo g 2 1 u = lo g 2 1 u 2 , 3 lo g 2 1 v = lo g 2 1 v 3 .
Use the quotient rule to combine the terms inside the parentheses: lo g 2 1 u 2 − lo g 2 1 v 3 = lo g 2 1 v 3 u 2 .
Use the product rule to combine the remaining terms: lo g 2 1 w 4 + lo g 2 1 v 3 u 2 = lo g 2 1 ( w 4 ⋅ v 3 u 2 ) .
Simplify the expression: lo g 2 1 ( v 3 w 4 u 2 ) . The final answer is lo g 2 1 ( v 3 w 4 u 2 ) .
Explanation
Understanding the Problem We are given the expression 4 lo g 2 1 w + ( 2 lo g 2 1 u − 3 lo g 2 1 v ) and we want to write it as a single logarithm. We will use the properties of logarithms to achieve this.
Applying the Power Rule First, we use the power rule of logarithms, which states that a lo g b x = lo g b x a . Applying this rule to each term, we get:
4 lo g 2 1 w = lo g 2 1 w 4
2 lo g 2 1 u = lo g 2 1 u 2
3 lo g 2 1 v = lo g 2 1 v 3
Substituting Back Now we substitute these back into the original expression:
lo g 2 1 w 4 + ( lo g 2 1 u 2 − lo g 2 1 v 3 )
Applying the Quotient Rule Next, we use the quotient rule of logarithms, which states that lo g b x − lo g b y = lo g b y x . Applying this to the terms inside the parentheses, we have:
lo g 2 1 u 2 − lo g 2 1 v 3 = lo g 2 1 v 3 u 2
Substituting Again Substituting this back into the expression, we get:
lo g 2 1 w 4 + lo g 2 1 v 3 u 2
Applying the Product Rule Finally, we use the product rule of logarithms, which states that lo g b x + lo g b y = lo g b ( x y ) . Applying this to the remaining terms, we have:
lo g 2 1 w 4 + lo g 2 1 v 3 u 2 = lo g 2 1 ( w 4 ⋅ v 3 u 2 ) = lo g 2 1 ( v 3 w 4 u 2 )
Final Answer Therefore, the expression 4 lo g 2 1 w + ( 2 lo g 2 1 u − 3 lo g 2 1 v ) written as a single logarithm is lo g 2 1 ( v 3 w 4 u 2 ) .
Examples
Logarithms are used extensively in various fields, such as calculating the magnitude of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth in biology. In finance, logarithms can help analyze investment growth rates and calculate the time it takes for an investment to double at a certain interest rate. Understanding how to combine logarithmic expressions is crucial for simplifying complex calculations and making informed decisions in these areas.
