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 (1)
Rewrite the equation using the change of base formula.
Simplify the equation using logarithm properties.
Solve the resulting quadratic equation.
Check for extraneous solutions and choose the valid solution: 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, which means we need to find the x-value where the two graphs intersect. We can solve this equation algebraically and then compare our solution to 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: 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: 2 lo g 2 ( 3 x − 1 ) = lo g 2 ( x + 8 )
Power Rule Use the power rule of logarithms: 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 ⟹ 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 = 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 Validity Check the validity of the solutions. Since we have lo g 2 ( 3 x − 1 ) , we require 0"> 3 x − 1 > 0 , so \frac{1}{3}"> x > 3 1 . Since we have lo g 4 ( x + 8 ) , we require 0"> x + 8 > 0 , so -8"> x > − 8 . Therefore, we need \frac{1}{3}"> x > 3 1 .
Extraneous Solution The solution x 1 ≈ 1.35 satisfies \frac{1}{3}"> x > 3 1 . The solution x 2 ≈ − 0.57 does not satisfy \frac{1}{3}"> x > 3 1 , so it is extraneous.
Final Answer The approximate solution is therefore 1.35, which is closest to 1.4 among the given options.
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 in these real-world scenarios.
