Write the equation of the circle in general form. Show your work.

Find x?<br> In 3x - In(x - 4) = ln(2x - 1) +ln3

earnstyle [38]

Answer:

$x = \displaystyle \frac{5 + \sqrt{17}}{2}$ .

Step-by-step explanation:

Because $3\, x$ is found in the input to a logarithm function in the original equation, it must be true that $3\, x > 0$ . Therefore, $x > 0$ .

Similarly, because $(x - 4)$ and $(2\, x - 1)$ are two other inputs to the logarithm function in the original equation, they should also be positive. Therefore, $x > 4$ .

Let $a$ and $b$ represent two positive numbers (that is: $a > 0$ and $b > 0$ .) The following are two properties of logarithm:

$\displaystyle \ln (a) + \ln(b) = \ln\left(a \cdot b\right)$ .

$\displaystyle \ln (a) - \ln(b) = \ln\left(\frac{a}{b}\right)$ .

Apply these two properties to rewrite the original equation.

Left-hand side of this equation:

$\begin{aligned}&\ln(3\, x) - \ln(x - 4)= \ln\left(\frac{3\, x}{x -4}\right)\end{aligned}$

Right-hand side of this equation:

$\ln(2\, x- 1) + \ln(3) = \ln\left(3 \left(2\, x - 1\right)\right)$ .

Equate these two expressions:

$\begin{aligned}\ln\left(\frac{3\, x}{x -4}\right) = \ln(3(2\, x - 1))\end{aligned}$ .

The natural logarithm function $\ln$ is one-to-one for all positive inputs. Therefore, for the equality $\begin{aligned}\ln\left(\frac{3\, x}{x -4}\right) = \ln(3(2\, x - 1))\end{aligned}$ to hold, the two inputs to the logarithm function have to be equal and positive. That is:

$\displaystyle \frac{3\ x}{x - 4} = 3\, (2\, x - 1)$ .

Simplify and solve this equation for $x$ :

$x^2 - 5\, x + 2 = 0$ .

There are two real (but not rational) solutions to this quadratic equation: $\displaystyle \frac{5 + \sqrt{17}}{2}$ and $\displaystyle \frac{5 - \sqrt{17}}{2}$ .

However, the second solution, $\displaystyle \frac{5 - \sqrt{17}}{2}$ , is not suitable. The reason is that if $x = \displaystyle \frac{5 - \sqrt{17}}{2}$ , then $(x - 4)$ , one of the inputs to the logarithm function in the original equation, would be smaller than zero. That is not acceptable because the inputs to logarithm functions should be greater than zero.

The only solution that satisfies the requirements would be $\displaystyle \frac{5 + \sqrt{17}}{2}$ .

Therefore, $x = \displaystyle \frac{5 + \sqrt{17}}{2}$ .

7 0

2 years ago

Describe the process you would use to explain to your parents (or other significant adults in your life) how you could calculate

9966 [12]

Answer:

Sum of interior angles of 12 sided object = 1800°

Step-by-step explanation:

Formula to calculate the sum of interior angles of a polygon is,

Sum of the interior angles of a polygon = (n - 2) × 180°

Where n = number of sides of the polygon.

If number of sides of a polygon are 12,

For n = 12,

Sum of interior angles = (12 - 2) × 180°

= 10 × 180°

= 1800°

Therefore, sum of interior angles of a 12 sided polygon will be 1800°.

5 0

2 years ago

About what percent of the village women took between 48 and 56 minutes to collect one container of water?

lisov135 [29]

The correct answer is 50%.

Approximately the middle 50% of data is contained inside the box of a box plot.

6 0

3 years ago

Read 3 more answers

I’ll give brainliest:)

Natasha2012 [34]

Answer:

ok ok ok ok ok ok okok ok ok

6 0

3 years ago

Read 2 more answers

Which of the following equations have the correct solution select all that apply

KATRIN_1 [288]

The First One
The Third One
The Forth One

The others result in a lower number than what they are supposed to be. The 3 correct ones end with the exact number of the equations solution number.

7 0

2 years ago