Express 50 billion in scientific notation

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

3 years ago

What is the area of the frame around the picture?

larisa [96]

12 x 6s - 8 x 4s = 72s - 32s = 40s

3 0

3 years ago

Read 2 more answers

Shawn drew a rectangle that had a width of 4.2 inches and a length of 8.1 inches. Find the perimeter of Shawn's rectangle

MArishka [77]

Answer: 24.6 inches

Step-by-step explanation: Since it's a rectangle, and not a rectangular prism, that means it's 2D. Length is the long sides on the top and bottom, and width is the shorter sides on it's left and right. So since there's two of each, just do 8.1 + 8.1 + 4.2 + 4.2 and you have your answer!

3 0

2 years ago

Vector v has a direction of (-4,27) find the direction angle for v

mixas84 [53]

$\bf \begin{array}{lclll}
\ \textless \ &-4&,&27\ \textgreater \ \\
&\uparrow &&\uparrow \\
&a&&b
\end{array}\qquad tan(\theta)=\cfrac{b}{a}\implies \measuredangle \theta=tan^{-1}\left( \frac{b}{a} \right)
\\\\\\
\measuredangle \theta=tan^{-1}\left( \frac{27}{-4} \right)$

8 0

3 years ago

Paula owns a shoe store in Los Angeles. Whenever she orders a new style of shoe, she orders 5 pairs of that style in each of the

suter [353]

She orders 5 pair in 5 different sizes:

5 x 5 = 25 pair

She orders 3 pair in 3 different sizes:

3 x 3 = 9 pair

Total pairs = 25 + 9 = 34 pair.

7 0

2 years ago