Question

Solve the logarithmic equation for x. (enter your answers as a comma-separated list. round your answer to four decimal places.) ln(x − 2) + ln(x + 3) = 1

Answer 1

To solve the equation ln(x − 2) + ln(x + 3) = 1, combine the logs into a single natural logarithmic expression, exponentiate to remove the log, and then solve the resulting quadratic equation using the quadratic formula, taking into account the domain limitations of the logarithmic functions.

Step-by-step explanation:

To solve the logarithmic equation ln(x − 2) + ln(x + 3) = 1, we can use the logarithm property that states the logarithm of a product of two numbers is the sum of the logarithms of the two numbers. Thus, we can combine the two logarithms into one by multiplying the two expressions inside the logarithms:

• ln(x − 2) + ln(x + 3) = ln((x − 2) × (x + 3))
• ln((x − 2)(x + 3)) = 1

By taking the natural logarithm of both sides of the equation, we can solve for x:

1. eln((x − 2)(x + 3)) = e1
2. (x − 2)(x + 3) = e
3. x2 + x − 6 = e
4. x2 + x − e − 6 = 0

Now we have a quadratic equation in standard form. We can solve this equation by using the quadratic formula:

• x = −(b)/(2a) ± √(b2 − 4ac)/(2a)
• x = −(1)/(2) ± √(12 − 4(1)(−e − 6))/(2)
• x = −(1)/(2) ± √(1 + 4e + 24)/(2)

The solutions for x are the values that satisfy the original equation, considering the domain of the natural logarithm function, which only admits positive input values. Therefore, we need to substitute the calculated x values back into the original equation to check if they make sense.

Remember to round your answer to four decimal places if you find valid solutions for x.

Answer 2

The two logs in ln(x − 2) + ln(x + 3) = 1 can be combined into one:


ln(x − 2) + ln(x + 3) = ln[ (x-2) * (x+3 ) ] = 1

Then (x-2)(x+3) = e^1 = e

Perform the indicated multiplication.

x^2 + 3x - 2x - 6 = e

1x^2 + 1x - (6+e) = 0

You can use the Quadratic Formula to find the roots here.

Note that a=1, b=1 and c = (6+e).

The discriminant is b^2 - 4ac, or 1^2 - 4(1)(6+e).

Unfortunately, the discriminant is negative, indicating that you'll have two complex roots.