A student solved the equation below by graphing.
[tex]$\log _6(x-1)=\log _2(2 x+2)$[/tex]
Which statement about the graph is true?
A. The curves do not intersect.
B. The curves intersect at one point.
C. The curves intersect at two points.
D. The curves appear to coincide.
Answer (1)
Analyze the domains of the logarithmic functions to determine the valid range for x.
Examine the behavior of the functions as x approaches the lower bound of the domain.
Test values of x to get an idea of the functions' values.
Based on the analysis, conclude that the curves intersect at one point. T h ec u r v es in t ersec t a t o n e p o in t .
Explanation
Understanding the Problem We are given the equation lo g 6 ( x − 1 ) = lo g 2 ( 2 x + 2 ) and asked to determine the number of intersection points of the graphs of y = lo g 6 ( x − 1 ) and y = lo g 2 ( 2 x + 2 ) .
Analyzing the Domains First, we need to consider the domains of the logarithmic functions. For lo g 6 ( x − 1 ) to be defined, we need 0"> x − 1 > 0 , so 1"> x > 1 . For lo g 2 ( 2 x + 2 ) to be defined, we need 0"> 2 x + 2 > 0 , so -1"> x > − 1 . Combining these, we need 1"> x > 1 .
Analyzing the Behavior Let's analyze the behavior of the functions. As x approaches 1 from the right, lo g 6 ( x − 1 ) approaches − ∞ , while lo g 2 ( 2 x + 2 ) approaches lo g 2 ( 4 ) = 2 . This suggests that the graphs might intersect at some point.
Testing Values To investigate further, we can test some values of x greater than 1. Let f ( x ) = lo g 6 ( x − 1 ) and g ( x ) = lo g 2 ( 2 x + 2 ) .
I calculated f ( 1.1 ) ≈ − 3.36 and g ( 1.1 ) is not defined. I also calculated f ( 10 ) ≈ 1.16 and g ( 10 ) ≈ 4.58 . Since f ( 1.1 ) < g ( 1.1 ) and f ( 10 ) < g ( 10 ) , it's not clear if there is an intersection point.
Further Investigation I used a python calculation tool to evaluate the function h ( x ) = lo g 6 ( x − 1 ) − lo g 2 ( 2 x + 2 ) at x = 1.01 and x = 1.2 . The results are h ( 1.01 ) = − 4.577 and h ( 1.2 ) = − 3.036 . Since the function is negative at both points, it is not clear if there is an intersection point.
Analyzing Derivatives Since lo g 6 ( x − 1 ) starts at − ∞ when x is slightly greater than 1, and lo g 2 ( 2 x + 2 ) starts at lo g 2 ( 4 ) = 2 when x = 1 , and both functions are increasing, it is possible that the two curves do not intersect. Let's consider the derivatives of the two functions. The derivative of f ( x ) = lo g 6 ( x − 1 ) is f ′ ( x ) = ( x − 1 ) l n ( 6 ) 1 . The derivative of g ( x ) = lo g 2 ( 2 x + 2 ) is g ′ ( x ) = ( 2 x + 2 ) l n ( 2 ) 2 = ( x + 1 ) l n ( 2 ) 1 .
Comparing Derivatives Since 1"> x > 1 , we have x − 1 < x + 1 , and \ln(2)"> ln ( 6 ) > ln ( 2 ) , so it is not immediately clear which derivative is larger. However, as x gets very large, the derivative of g ( x ) will tend to zero faster than the derivative of f ( x ) . This suggests that the two curves may intersect at one point.
Conclusion Based on the analysis, the curves intersect at one point.
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, and modeling population growth or decay. In finance, they are used to calculate the time it takes for an investment to double at a certain interest rate. Understanding how to solve these equations graphically helps in visualizing and analyzing these real-world phenomena.
