Solve each equation. ln(2x+1)+ln(x−3)−2 ln x=0

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Hello. Today we're going to be solving the given equation for X. Now the given equation is the natural log of four X plus three plus the natural log of x minus one minus two times the natural log of X Is equal to zero. Now the first thing we're gonna want to do is go ahead and add two times the natural log of X to both sides of the equation. And doing so is going to give us the natural log of four, X plus three plus the natural log of x minus one is equal to two times the natural log of X. Now we're gonna want to rewrite the left hand side using the addition rule for the natural logs. And that states that if you have the natural log of A plus the natural log of B, Then that is equal to the natural log of eight times b. And on the right hand side we want to go ahead and get rid of this too. And we can do that using the exponential rule for natural logs. And that states that if you have the natural log of X raised to the power of A, you can bring forward the A in front of the natural log as a product or in other words the natural log of X raised to the power of A is equal to a times the natural log of X. But in this case we're going to be going from right to left because they're equal both ways. So rewriting the left hand side. Using the addition rule is going to give us the natural log of the quantities four X plus three times x minus one. And that is going to be equal to the natural log of X squared Now, since both sides of the equation have natural logs, we can just go ahead and equate the inside arguments to each other. And doing this is going to give us the quantity four X plus three Times X -1 is equal to x squared. Now on the left hand side we're gonna want to open up these parentheses and in order to do that, we're going to go ahead and foil the left hand side, foiling the left hand side is going to give us four X squared minus x minus three. And that is going to be equal to X squared. And since we want to go ahead and sulfur X, we're going to set this equation equal to zero by subtracting X squared to both sides of the equation. And doing this is going to give us three, X squared minus x minus three is equal to zero. Now, unfortunately, this is not a fact herbal trin Amiel. So we're going to need to solve this using the quadratic equation. So let's go ahead and identify our coefficients A is equal to three. B is equal to negative one and C is equal to negative three. And this is where X is equal to negative B plus or minus the square root of B squared minus four times a, times C All over two times A. So we're just gonna go ahead and directly plug in our coefficients to the quadratic formula. So X is going to equal to negative B which is negative negative one or positive one plus or minus the square root of negative one squared minus four times A which is three times C which is negative three, all over two times A which is two times three. Okay, now we need to go ahead and simplify negative one squared is going to give us one and negative four times three times negative three is going to give us 36 So X is equal to one plus or minus the square root of one plus 36. All over two times three which is six and one plus 36 is going to give us 37 so X is equal to one plus or minus the square root of 37/6. So this gives us two values for X to work with X is equal to one plus the square root of 37/ and X is equal to one minus the square root of 37/6. Now this is a really complicated number to work with. So let's go ahead and rewrite them in decimal form. The positive form is going to give us X is equal to 1.1805 and X is equal to negative 0.8471. And so since we've worked with natural logs, we need to go ahead and verify that both of these values work with it. So we're going to take our given values of X and directly plug it into our original equation. So starting with X is equal to 1.1805. Plugging that into the original equation is going to give us natural log of four times 1.1805 plus three plus The natural log of 1.1805 - -2 times the natural log of 1.1805 is going equal to zero. Now if you simplify all the natural logs and add all the values together, what this leaves you with is zero is equal to zero, which is a true statement. So that means that X is equal to 1.1805 is going to be a solution for X. But now we need to check the other value of X. So when X is equal to negative 0.8471, Plugging this into the original equation is going to give us the natural log of four times negative 0. plus three Plus the natural log of negative 0.847, -1 -2 times the natural log of negative 0.8471 is equal to zero. But notice one thing about this, this value of the natural log has a negative number inside the natural log. So this is going to produce an undefined value. And in this value you have a negative minus a negative, which is going to produce a negative number and a negative inside a natural log is going to produce an undefined value. So overall we're gonna have undefined values for the negative value of X. So this cannot be a solution for X, which means that there is only one solution for X. X is equal to 1.1805. And with that being said, the answer to this problem is going to be a So I hope this video helps you in understanding how to solve this type of equation and I'll go ahead and see you all in the next video.

Solve each equation. ln(2x+1)+ln(x−3)−2 ln x=0

Key Concepts

Properties of Logarithms

Domain of Logarithmic Functions

Solving Logarithmic Equations

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