Kim solved the equation below by graphing a system of equations.
[tex]\log _2(3 x-1)=\log _4(x+8)[/tex]
What is the approximate solution to the equation?
A. 0.6
B. 0.9
C. 1.4
D. 1.6
Answer (2)
Rewrite the equation using the change of base formula: lo g 4 ( x + 8 ) = 2 1 lo g 2 ( x + 8 ) .
Simplify the equation to ( 3 x − 1 ) 2 = x + 8 .
Solve the quadratic equation 9 x 2 − 7 x − 7 = 0 using the quadratic formula.
Choose the valid solution x ≈ 1.35 that satisfies the domain \frac{1}{3}"> x > 3 1 , which is closest to 1.4 .
Explanation
Understanding the Problem We are given the equation lo g 2 ( 3 x − 1 ) = lo g 4 ( x + 8 ) and asked to find the approximate solution. Kim solved this by graphing a system of equations. We will solve it algebraically and then choose the closest answer from the given options.
Change of Base First, we rewrite the equation using the change of base formula. We change the base of the right-hand side to base 2. Recall that lo g a b = l o g c a l o g c b . Thus, lo g 4 ( x + 8 ) = l o g 2 4 l o g 2 ( x + 8 ) = 2 l o g 2 ( x + 8 ) .
Rewriting the Equation The equation becomes lo g 2 ( 3 x − 1 ) = 2 1 lo g 2 ( x + 8 ) .
Multiplying by 2 Multiply both sides by 2 to get 2 lo g 2 ( 3 x − 1 ) = lo g 2 ( x + 8 ) .
Power Rule of Logarithms Use the power rule of logarithms to rewrite the left side: lo g 2 ( 3 x − 1 ) 2 = lo g 2 ( x + 8 ) .
Equating Arguments Since the logarithms are equal, the arguments must be equal: ( 3 x − 1 ) 2 = x + 8 .
Expanding and Simplifying Expand and simplify the quadratic equation: 9 x 2 − 6 x + 1 = x + 8 , which simplifies to 9 x 2 − 7 x − 7 = 0 .
Quadratic Formula Solve the quadratic equation for x using the quadratic formula: x = 2 a − b ± b 2 − 4 a c , where a = 9 , b = − 7 , and c = − 7 . Thus, x = 2 ( 9 ) 7 ± ( − 7 ) 2 − 4 ( 9 ) ( − 7 ) = 18 7 ± 49 + 252 = 18 7 ± 301 .
Calculating Roots Calculate the two possible values of x: x 1 = 18 7 + 301 ≈ 18 7 + 17.35 ≈ 1.35 and x 2 = 18 7 − 301 ≈ 18 7 − 17.35 ≈ − 0.57 .
Checking the Domain Since the domain of lo g 2 ( 3 x − 1 ) requires 0"> 3 x − 1 > 0 , which means \frac{1}{3}"> x > 3 1 , and the domain of lo g 4 ( x + 8 ) requires 0"> x + 8 > 0 , which means -8"> x > − 8 , we must have \frac{1}{3}"> x > 3 1 . Therefore, x 2 is not a valid solution.
Final Answer The approximate solution is x 1 ≈ 1.35 , which is closest to 1.4 among the given options. Therefore, the approximate solution to the equation is 1.4.
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 logarithmic equations helps in making informed decisions and predictions in these real-world scenarios.
The approximate solution to the equation lo g 2 ( 3 x − 1 ) = lo g 4 ( x + 8 ) is found to be approximately 1.35. This value is closest to 1.4, making option C the correct choice.
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