If the equation below is solved by graphing, which statement is true?
[tex]$\log (6 x+10)=\log _{\frac{1}{2}} x$[/tex]
A. The curves intersect at approximately [tex]$x=0.46$[/tex].
B. The curves intersect at approximately [tex]$x=0.75$[/tex].
C. The curves intersect at approximately [tex]$x=1.11$[/tex].
D. The curves intersect at approximately [tex]$x=3.07$[/tex].
Answer (1)
Define the function f ( x ) = lo g ( 6 x + 10 ) − lo g 2 1 ( x ) .
Evaluate f ( x ) for each given x value: 0.46, 0.75, 1.11, and 3.07.
Determine which f ( x ) is closest to 0.
The closest value is f ( 0.46 ) ≈ − 0.0144 , so the answer is 0.46 .
Explanation
Understanding the Problem We are given the equation lo g ( 6 x + 10 ) = lo g 2 1 x and four possible solutions for x . We need to determine which of the given x values is the approximate solution to the equation. To do this, we can define a function f ( x ) = lo g ( 6 x + 10 ) − lo g 2 1 ( x ) and test each of the given x values to see which one is closest to being a solution, i.e., which one makes f ( x ) closest to 0.
Testing the Given x-values Let's evaluate f ( x ) for each of the given x values:
For x = 0.46 :
f ( 0.46 ) = lo g ( 6 ( 0.46 ) + 10 ) − lo g 2 1 ( 0.46 ) = lo g ( 2.76 + 10 ) − lo g 2 1 ( 0.46 ) = lo g ( 12.76 ) − lo g 2 1 ( 0.46 ) Using a calculator, we find that f ( 0.46 ) ≈ − 0.0144 .
For x = 0.75 :
f ( 0.75 ) = lo g ( 6 ( 0.75 ) + 10 ) − lo g 2 1 ( 0.75 ) = lo g ( 4.5 + 10 ) − lo g 2 1 ( 0.75 ) = lo g ( 14.5 ) − lo g 2 1 ( 0.75 ) Using a calculator, we find that f ( 0.75 ) ≈ 0.7463 .
For x = 1.11 :
f ( 1.11 ) = lo g ( 6 ( 1.11 ) + 10 ) − lo g 2 1 ( 1.11 ) = lo g ( 6.66 + 10 ) − lo g 2 1 ( 1.11 ) = lo g ( 16.66 ) − lo g 2 1 ( 1.11 ) Using a calculator, we find that f ( 1.11 ) ≈ 1.3722 .
For x = 3.07 :
f ( 3.07 ) = lo g ( 6 ( 3.07 ) + 10 ) − lo g 2 1 ( 3.07 ) = lo g ( 18.42 + 10 ) − lo g 2 1 ( 3.07 ) = lo g ( 28.42 ) − lo g 2 1 ( 3.07 ) Using a calculator, we find that f ( 3.07 ) ≈ 3.0719 .
Finding the Closest Solution The x value that results in f ( x ) closest to 0 is the approximate solution. Comparing the values of f ( x ) for each x , we have:
f ( 0.46 ) ≈ − 0.0144 f ( 0.75 ) ≈ 0.7463 f ( 1.11 ) ≈ 1.3722 f ( 3.07 ) ≈ 3.0719
The value closest to 0 is f ( 0.46 ) ≈ − 0.0144 . Therefore, the curves intersect at approximately x = 0.46 .
Final Answer Therefore, the correct statement is: The curves intersect at approximately x = 0.46 .
Examples
Logarithmic equations are used in various fields, such as calculating the magnitude of earthquakes on the Richter scale, determining the acidity or alkalinity (pH) of a solution in chemistry, and modeling population growth or decay in biology. In finance, they can be used to calculate the time it takes for an investment to double at a certain interest rate. Understanding how to solve logarithmic equations helps in making informed decisions and predictions in these real-world scenarios.
