Question

Solve 1n x = 2 + 1n(x - 4). Give the exact answer then use a calculator to find an approximate answer.

Answer 1

To solve ln(x) = 2 + ln(x - 4), properties of logarithms are used to combine the logs and isolate x, resulting in a quadratic equation. Using a calculator, the approximate solution for x is 0.00139, ignoring the negative solution as it's not relevant in this context.

Step-by-step explanation:

To solve the equation 1n x = 2 + 1n(x - 4), you can use properties of logarithms to simplify and solve for x. The equation involves the natural logarithm, often written as 'ln'.

1. First, recognize that '1n' should be 'ln,' which stands for natural logarithm. So the equation should be ln(x) = 2 + ln(x - 4).
2. Next, we can use the property that ln(a) - ln(b) = ln(a/b) to combine the logarithms on the right side: ln(x) - ln(x - 4) = 2.
3. This simplifies to ln(x/(x - 4)) = 2. Raise e to both sides of the equation to get rid of the ln, leading to x/(x - 4) = e^2.
4. Solve this equation for x, yielding x = e^2(x - 4).
5. Expand and rearrange: x = e^2x - 4e^2. Then, 0 = e^2x - x - 4e^2.
6. This is now a quadratic equation in terms of e^2x, so let u = e^2x, giving us 0 = u - e^-2u - 4e^2.
7. Solve this quadratic equation to find the values of u, and then calculate x.

When solving using a calculator, we get that the solution for x is approximately 0.00139, not taking into account the negative value as it doesn't have relevance in this logarithmic context.

Answer 2

To solve the logarithmic equation ln x = 2 + ln(x - 4), subtract ln(x - 4) from both sides, apply logarithmic properties, and rearrange the equation to solve for x, using a calculator for approximation.

Step-by-step explanation:

To solve the equation 1n x = 2 + 1n(x - 4), we first recognize that 1n is a common typo for 'ln', which represents the natural logarithm function. Our aim is to isolate variables and solve for x. Here's how we do it step-by-step:

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However, it is also important to check whether the solution makes sense. For example, a negative x would not be valid when plugged back into ln(x - 4) since the logarithm of a negative number is undefined.