Question

If p(x) = 2ln(x)-1 and m(x) =ln(x+6), then what is the solution for p(x) = m(x)

Answer 1

x ≈ 3.40295

Explanation:

We want to find the value(s) of x that satisfy the equation p(x) = m(x), where p(x) = 2ln(x) - 1 and m(x) = ln(x + 6).

So, we have:

2ln(x) - 1 = ln(x + 6)

First, let's simplify the equation by using the properties of logarithms:

ln(x^2) - ln(e) = ln(x + 6)

ln(x^2/e) = ln(x + 6)

Now we can eliminate the natural logarithm by taking the exponential of both sides:

x^2/e = x + 6

x^2 - ex - 6e = 0

We can solve this quadratic equation using the quadratic formula:

x = [-(-e) ± sqrt((-e)^2 - 4(1)(-6e))] / (2(1))

Simplifying:

x = [e ± sqrt(e^2 + 24e)] / 2

Therefore, the solutions for p(x) = m(x) are:

x = [e + sqrt(e^2 + 24e)] / 2 or x = [e - sqrt(e^2 + 24e)] / 2

Note that since the argument of the natural logarithm must be positive, we must discard the negative solution, which is not valid. So the only solution is:

x = [e + sqrt(e^2 + 24e)] / 2

We can approximate this value using a calculator or computer software. For example, if we use e = 2.71828, we get:

x ≈ 3.40295

Therefore, the solution for p(x) = m(x) is x ≈ 3.40295.

Answer 2

Explanation:

To find the solution for p(x) = m(x), we need to set the two expressions equal to each other and solve for x:

2ln(x) - 1 = ln(x+6)

First, we can simplify the left side by using the logarithmic identity that states ln(a^b) = b ln(a):

ln(x^2) - 1 = ln(x+6)

Next, we can eliminate the natural logarithm by exponentiating both sides with the base e:

e^[ln(x^2) - 1] = e^[ln(x+6)]

e^[ln(x^2)] * e^[-1] = x + 6

x^2 * 1/e = x + 6

Multiplying both sides by e gives:

x^2 = e(x + 6)

Expanding the right side gives:

x^2 = ex + 6e

Rearranging gives:

x^2 - ex - 6e = 0

This is a quadratic equation in x. We can solve for x using the quadratic formula:

x = [e ± sqrt(e^2 + 4*6e)]/2

Simplifying gives:

x = [e ± sqrt(e^2 + 24e)]/2

Therefore, the solution for p(x) = m(x) is x = [e ± sqrt(e^2 + 24e)]/2.