What is the solution to the equation below? log6 4x 2 – log6 x=2
The Correct Answer and Explanation is:
To solve the equation log6(4×2)−log6(x)=2\log_6 (4x^2) – \log_6 (x) = 2log6(4×2)−log6(x)=2, follow these steps:
Step 1: Simplify the logarithmic expression
Using the logarithmic property logb(A)−logb(B)=logb(AB)\log_b (A) – \log_b (B) = \log_b \left(\frac{A}{B}\right)logb(A)−logb(B)=logb(BA), rewrite the left-hand side:log6(4×2)−log6(x)=log6(4x2x)\log_6 (4x^2) – \log_6 (x) = \log_6 \left(\frac{4x^2}{x}\right)log6(4×2)−log6(x)=log6(x4x2)
Simplify the fraction 4x2x\frac{4x^2}{x}x4x2:log6(4x2x)=log6(4x)\log_6 \left(\frac{4x^2}{x}\right) = \log_6 (4x)log6(x4x2)=log6(4x)
The equation now becomes:log6(4x)=2\log_6 (4x) = 2log6(4x)=2
Step 2: Rewrite the logarithmic equation in exponential form
Using the definition of a logarithm logb(A)=C ⟺ bC=A\log_b (A) = C \iff b^C = Alogb(A)=C⟺bC=A, rewrite log6(4x)=2\log_6 (4x) = 2log6(4x)=2:62=4×6^2 = 4×62=4x
Simplify 626^262:36=4×36 = 4×36=4x
Step 3: Solve for xxx
Divide both sides of the equation by 4:x=364=9x = \frac{36}{4} = 9x=436=9
Step 4: Verify the solution
Substitute x=9x = 9x=9 back into the original equation to confirm:log6(4(9)2)−log6(9)=log6(4⋅81)−log6(9)=log6(324)−log6(9)\log_6 (4(9)^2) – \log_6 (9) = \log_6 (4 \cdot 81) – \log_6 (9) = \log_6 (324) – \log_6 (9)log6(4(9)2)−log6(9)=log6(4⋅81)−log6(9)=log6(324)−log6(9)
Using the logarithmic property:log6(324)−log6(9)=log6(3249)=log6(36)\log_6 (324) – \log_6 (9) = \log_6 \left(\frac{324}{9}\right) = \log_6 (36)log6(324)−log6(9)=log6(9324)=log6(36)
Since log6(36)=2\log_6 (36) = 2log6(36)=2, the solution is correct.
Final Answer:
x=9x = 9x=9
Explanation (300 words)
This problem involves logarithmic properties to simplify and solve an equation. The key idea is to apply the subtraction rule of logarithms, logb(A)−logb(B)=logb(AB)\log_b (A) – \log_b (B) = \log_b \left(\frac{A}{B}\right)logb(A)−logb(B)=logb(BA), which reduces two logarithms into one. Here, log6(4×2)−log6(x)\log_6 (4x^2) – \log_6 (x)log6(4×2)−log6(x) simplifies to log6(4x2x)=log6(4x)\log_6 \left(\frac{4x^2}{x}\right) = \log_6 (4x)log6(x4x2)=log6(4x).
After simplifying, the equation becomes log6(4x)=2\log_6 (4x) = 2log6(4x)=2. The next step is converting the logarithmic equation into its exponential form. By definition, logb(A)=C\log_b (A) = Clogb(A)=C is equivalent to bC=Ab^C = AbC=A. Using this, log6(4x)=2\log_6 (4x) = 2log6(4x)=2 translates to 62=4×6^2 = 4×62=4x, or 36=4×36 = 4×36=4x. Solving for xxx gives x=9x = 9x=9.
Verification confirms the solution: substituting x=9x = 9x=9 back into the original equation produces a true statement, proving the result is correct. This approach showcases how logarithmic rules and exponential transformations simplify complex equations.