Question

What is the solution to the equation? lnx−ln(2x−3)=ln3

Answer 1

The solution to the equation ln(x) - ln(2x - 3) = ln(3) is x = 1.8, found by applying the property of logarithms that allows us to combine the log terms and set the arguments equal to each other.

Step-by-step explanation:

To solve the equation ln(x) - ln(2x - 3) = ln(3), we can use the property of logarithms that the logarithm of the number resulting from the division of two numbers is the difference between the logarithms of the two numbers. Applying this property, we rewrite the equation as ln(x/(2x - 3)) = ln(3).

Since logarithms of the same base are equivalent only if their arguments are equivalent, we can set the arguments of the natural logarithms equal to each other:

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• x / (2x - 3) = 3

Now we solve for x:

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• x = 3(2x - 3)
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• x = 6x - 9
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• 5x = 9
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• x = 9/5
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• x = 1.8

Therefore, the solution to the equation is x = 1.8.

Note that we must also check the solution to ensure that it does not result in the logarithm of a negative number, as that would be outside the domain of the logarithmic function. Since 2(1.8) - 3 = 0.6, which is positive, our solution is valid.