solve: Ln(x+2)-Ln(4x+3)=Ln(1/x)

I think this is a Log of Quotient

ln( x+2)/(4x+3) ) = ln (/x)

(x+2)/(4x+3) = 1/x
x^2 + 2x = 4x+3
x^2 - 2x - 3 = 0
(x-3)(x+1) = 0
x = 3 or x = -1

but if x = -1 , ln(4x+3) is undefined, so

x = 3

Ah, the joy of logarithms! Let's solve this logarithmic puzzle, shall we?

We have the equation Ln(x+2) - Ln(4x+3) = Ln(1/x). Since we're dealing with natural logarithms here, we can use the log of quotient rule, which states:

Ln(a) - Ln(b) = Ln(a/b)

So, applying this rule to our equation, we get:

Ln((x+2)/(4x+3)) = Ln(1/x)

Now, using the wonderful property of logarithms that Ln(a) = Ln(b) if and only if a = b, we can drop the Ln on both sides:

(x+2)/(4x+3) = 1/x

Now, let's eliminate the fractions by cross-multiplying:

x(x+2) = (4x+3)(1)

Expanding both sides, we get:

x^2 + 2x = 4x + 3

Rearranging terms, we have:

x^2 + 2x - 4x - 3 = 0

Simplifying further, we get:

x^2 - 2x - 3 = 0

Now, it's time for a little humor break! Did you hear about the mathematician who was afraid of negative numbers? He will stop at nothing to avoid them! 🤣

Okay, back to business. To solve our quadratic equation, we can either factorize or use the quadratic formula. Let's opt for the quadratic formula:

x = (-b ± √(b^2 - 4ac))/(2a)

Plugging in our values a = 1, b = -2, and c = -3, we get:

x = (-(-2) ± √((-2)^2 - 4(1)(-3)))/(2(1))

Simplifying further, we have:

x = (2 ± √(4 + 12))/2

x = (2 ± √16)/2

x = (2 ± 4)/2

Now, let's consider the two possibilities:

1) x = (2 + 4)/2 = 6/2 = 3

2) x = (2 - 4)/2 = -2/2 = -1

So, we have two potential solutions: x = 3 and x = -1.

And there you have it! We've successfully solved the logarithmic equation. Keep up the great work, and remember, math can be fun too! 🎉

To solve the equation ln(x+2) - ln(4x+3) = ln(1/x), we can apply the properties of logarithms. One property states that ln(a) - ln(b) is equal to ln(a/b). Using this property, we can rewrite the equation as ln((x+2)/(4x+3)) = ln(1/x).

Next, we can apply the rule that if ln(x) = ln(y), then x = y. Using this rule, we can equate the two expressions inside the logarithms:

(x+2)/(4x+3) = 1/x

To solve for x, we can cross-multiply:

(x+2)x = (4x+3)

Now let's distribute:

x^2 + 2x = 4x + 3

Rearranging the equation:

x^2 - 2x - 4x + 3 = 0

Combining like terms:

x^2 - 6x + 3 = 0

Unfortunately, this equation cannot be easily factored, so we will need to use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our quadratic equation, a = 1, b = -6, and c = 3. Substituting these values into the quadratic formula:

x = (-(-6) ± √((-6)^2 - 4(1)(3))) / (2(1))

Simplifying further:

x = (6 ± √(36 - 12)) / 2

x = (6 ± √(24)) / 2

x = (6 ± √(4 * 6)) / 2

x = (6 ± 2√6) / 2

Simplifying again:

x = 3 ± √6

Therefore, the solutions to the equation ln(x+2) - ln(4x+3) = ln(1/x) are x = 3 + √6 and x = 3 - √6.

You're correct! The given equation, Ln(x+2) - Ln(4x+3) = Ln(1/x), can be solved using the properties of logarithms.

First, let's start by combining the logarithms using the logarithmic quotient rule, which states that Ln(a) - Ln(b) = Ln(a/b). Applying this rule to the equation gives us:

Ln((x+2)/(4x+3)) = Ln(1/x)

Now, we can remove the natural logarithm on both sides by taking the exponential of each side. The exponential function, e^x, is the inverse of the natural logarithm (Ln(x)), so applying it to both sides gives us:

e^(Ln((x+2)/(4x+3))) = e^(Ln(1/x))

Simplifying further, we have:

(x+2)/(4x+3) = 1/x

Now we can solve for x by cross-multiplying. Multiply both sides of the equation by x and (4x+3):

x(x+2) = (4x+3)

Expanding the equation gives us:

x^2 + 2x = 4x + 3

Rearranging the terms:

x^2 - 2x - 4x + 3 = 0

Combining like terms:

x^2 - 6x + 3 = 0

To solve this quadratic equation, we can use the quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / (2a)

In this case, a = 1, b = -6, and c = 3. Plugging these values into the quadratic formula, we get:

x = (-(-6) ± sqrt((-6)^2 - 4 * 1 * 3)) / (2 * 1)

Simplifying further:

x = (6 ± sqrt(36 - 12)) / 2

x = (6 ± sqrt(24)) / 2

x = (6 ± 2sqrt(6)) / 2

Now we can simplify the expression by canceling out the common factor:

x = 3 ± sqrt(6)

So the solutions to the equation Ln(x+2) - Ln(4x+3) = Ln(1/x) are x = 3 + sqrt(6) and x = 3 - sqrt(6).