Question

If ln(x) + ln(7 – x) = ln(12), then x belongs to... {3,4}. {3}. {4}. No correct answer.

Answer 1

Main Answer:

The correct solution for (x) in the given logarithmic equation ln(x)+ln(7ax)=ln(12) is x=4. The set of possible values for x is 4.

Step-by-step explanation:

To find the solution for x, we can use properties of logarithms. The given equation is \
`(ln(x) + ln(7 – x) = ln(12)\)`. According to the logarithmic property ln(a)+ln(b)=ln(a*b), we can combine the logarithms on the left side of the equation:
`\(ln(x * (7 – x)) = ln(12)\)`.

Now, we can set the arguments equal to each other:
`\(x * (7 – x) = 12\)`. Simplifying this quadratic equation yields x²-7x+12=0, which factors as (x-3)(x-4)=0. Therefore, the solutions for x are x = 3 and x = 4. However, since the natural logarithm function is only defined for positive values, x = 3 is extraneous, and the correct solution is x = 4. Hence, the set of possible values for x is {4}.

In summary, solving the logarithmic equation leads to a quadratic equation, and after considering the domain of the natural logarithm function, we find that the only valid solution is x = 4.