Solve:

\[
\ln(x - 3) = \ln(7x - 23) - \ln(x + 1)
\]

To solve the equation ln(x 3) = ln(7x 23) ln(x + 1), we can use the properties of logarithms. Specifically, we will use the property that states that the natural logarithm of a division of two numbers is the difference between the natural logarithms of the two numbers, and also recall that the exponential and natural logarithm functions are inverse functions.
First, we can combine the right side of the equation using this property: ln(a) - ln(b) = ln(a/b) , which gives us:
ln(x - 3) = ln((7x - 23)/(x + 1))
Since the natural logarithm function is one-to-one, we can equate the arguments of the logarithms when the logs are equal, which leaves us with:
x - 3 = (7x - 23)/(x + 1)
Multiplying both sides by (x + 1) to eliminate the denominator and then solving for x, we get:
(x - 3)(x + 1) = 7x - 23
x² - 2x - 3 = 7x - 23
x² - 9x + 20 = 0
Factoring the quadratic equation, we find:
(x - 5)(x - 4) = 0
Therefore, the solutions are x = 5 and x = 4. However, we must check these solutions in the original equation to ensure they do not make any logarithm undefined. After verifying, we find that x = 5 is the only solution that satisfies the original equation because x = 4 would lead to a negative argument in the first logarithm, which is not allowed.
So, the solution to the equation is x = 5.

Answered by WilliamClarkGable | 2024-06-18

The solution to the equation ln ( x − 3 ) = ln ( 7 x − 23 ) − ln ( x + 1 ) is found to be x = 5 , after simplifying using properties of logarithms and confirming the validity of the solution against the original equation. x = 4 did not yield a valid argument for the logarithm. Thus, the only solution is x = 5 .
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Answered by WilliamClarkGable | 2024-12-26