Solve the equation [tex]$\ln x+\ln (x-3)=\ln (6 x)$[/tex]
[tex]x=[/tex]
Answer (2)
Combine the logarithms on the left side using the property ln a + ln b = ln ( ab ) .
Equate the arguments of the logarithms since the logarithms are equal.
Solve the resulting quadratic equation.
Check the solutions to ensure they are valid within the domain of the logarithmic functions. The final answer is 9 .
Explanation
Understanding the Problem We are given the equation ln x + ln ( x − 3 ) = ln ( 6 x ) and we need to solve for x .
Combining Logarithms First, we use the logarithm property that states ln a + ln b = ln ( ab ) to combine the terms on the left side of the equation: ln ( x ( x − 3 )) = ln ( 6 x ) .
Equating Arguments Since the logarithms on both sides are equal, their arguments must be equal. Therefore, we can write: x ( x − 3 ) = 6 x .
Expanding the Equation Expanding the left side, we get: x 2 − 3 x = 6 x .
Rearranging to Quadratic Form Now, we rearrange the equation to form a quadratic equation: x 2 − 3 x − 6 x = 0 x 2 − 9 x = 0 .
Factoring the Quadratic Next, we factor the quadratic equation: x ( x − 9 ) = 0 .
Finding Potential Solutions This gives us two possible solutions for x :
x = 0 or x = 9 .
Checking for Validity However, we must check if these solutions are valid in the original equation. Recall that the logarithm function ln x is only defined for 0"> x > 0 . Also, we have the term ln ( x − 3 ) in the original equation, which requires 0"> x − 3 > 0 , or 3"> x > 3 .
If x = 0 , then ln x and ln ( x − 3 ) are not defined, so x = 0 is not a valid solution.
If x = 9 , then ln 9 , ln ( 9 − 3 ) = ln 6 , and ln ( 6 ⋅ 9 ) = ln 54 are all defined. Thus, x = 9 is a valid solution.
Therefore, the only valid solution is x = 9 .
Final Answer Thus, the solution to the equation ln x + ln ( x − 3 ) = ln ( 6 x ) is 9 .
Examples
Logarithmic equations are used in various fields such as finance, physics, and engineering. For example, in finance, they are used to model compound interest and calculate the time it takes for an investment to double. In physics, they appear in radioactive decay problems, where the logarithm helps determine the half-life of a substance. Understanding how to solve logarithmic equations is crucial for making informed decisions in these areas.
The solution to the equation ln x + ln ( x − 3 ) = ln ( 6 x ) is found by combining logarithms, equating their arguments, and solving a quadratic equation. After verifying potential solutions, we find that the only valid solution is x = 9 . Thus, the final answer is 9 .
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