Problem 4. (1 point)
[tex]\ln \left(r^6 s^7 \sqrt[9]{r^8 s^9}\right)[/tex]
is equal to
[tex]A \ln r+B \ln s[/tex]
where [tex]A=[/tex] $\square$ and where [tex]B=[/tex] $\square$
Note: In order to get credit for this problem all answers must be correct.
Answer (2)
Use the property ln ( x y ) = ln x + ln y to expand the logarithm.
Use the property ln ( x a ) = a ln x to simplify the expression.
Combine like terms to find the coefficients A and B.
The final answer is A = 9 62 and B = 8 .
Explanation
Understanding the Problem We are given the expression ln ( r 6 s 7 9 r 8 s 9 ) and we want to express it in the form A ln r + B ln s . Our goal is to find the values of A and B. To do this, we will use properties of logarithms to simplify the given expression.
Applying Logarithmic Properties - Step 1 First, we rewrite the expression using the property ln ( x y ) = ln x + ln y : ln ( r 6 s 7 9 r 8 s 9 ) = ln ( r 6 ) + ln ( s 7 ) + ln ( 9 r 8 s 9 )
Applying Logarithmic Properties - Step 2 Next, we use the property ln ( x a ) = a ln x : ln ( r 6 ) + ln ( s 7 ) + ln ( 9 r 8 s 9 ) = 6 ln r + 7 ln s + ln (( r 8 s 9 ) 9 1 )
Applying Logarithmic Properties - Step 3 Now, we use the property ( x y ) a = x a y a : 6 ln r + 7 ln s + ln (( r 8 s 9 ) 9 1 ) = 6 ln r + 7 ln s + ln ( r 9 8 s 9 9 )
Applying Logarithmic Properties - Step 4 Again, we use the property ln ( x a ) = a ln x : 6 ln r + 7 ln s + ln ( r 9 8 s 9 9 ) = 6 ln r + 7 ln s + 9 8 ln r + 9 9 ln s
Combining Like Terms Now, we combine the terms with ln r and ln s : 6 ln r + 7 ln s + 9 8 ln r + 9 9 ln s = ( 6 + 9 8 ) ln r + ( 7 + 1 ) ln s
Simplifying the Coefficients We simplify the coefficients: ( 6 + 9 8 ) ln r + ( 7 + 1 ) ln s = 9 62 ln r + 8 ln s
Finding the Values of A and B Thus, we have A = 9 62 and B = 8 . Since A = 9 62 ≈ 6.89 and we are asked if A = 10 , the answer is no. Since B = 8 and we are asked if B = 2 23 = 11.5 , the answer is no.
Final Answer Therefore, A = 9 62 and B = 8 .
Examples
Logarithms are incredibly useful in many fields, such as engineering, physics, and computer science. For example, in electrical engineering, logarithms are used to measure signal strength in decibels. The properties of logarithms, like those used in this problem, help simplify complex calculations and make them easier to understand and apply in real-world scenarios. Understanding how to manipulate logarithmic expressions allows engineers to analyze and design systems more efficiently.
We simplified the expression ln ( r 6 s 7 9 r 8 s 9 ) using logarithmic properties, resulting in A = 9 62 and B = 8 .
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