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First off, since both of them equal to y, then we can set the two equations to equal each other. We have

\(\log2 (2x) =\log4 (16 + x)\)

Applying the change of base thereom to the equation, we get

\(\log (2x) / \log2 = \log (16 + x) / \log 4 \\ \log (2x) / \log 2 = \log (16 + x) /\log 2^2 \\ \log (2x) / \log 2 = \log (16 + x) / [ 2 \log 2]\)

Now, we multiply through by log 2. 

\(\log (2x) =\log (16 + x) / 2 \\ 2\log (2x) = \log (16 + x) \\ \log (2x)^2 = \log (16 + x)\)

Now, wait a second. Notice that both logs have the same base now. 

This means there aren't any reprucussions we have to worry about. We simply just take the inside of the parenthesis. 

This implies that

\((2x)^2 = 16 + x \\ 4x^2 - x - 16 = 0\)

Taking the postiive value of x and using the quadratic equation, we get

The answer to the system of equations is \(x = [ 1 + \sqrt{257} ] / 8\)

First, let's make some observations about this question. 

All points 5 units away from point (2, 7) would make a circle with center (2, 7) and radius 5. 

The formula for that circle would look something like \((x-2)^2+(y-7)^2=25\)

Now, combining that with the second equation, and we get a system. 

Subsituting out y with the x values in the line equation, we get

\((x-2)^2+(5x-35)^2=25\)

Simplifying, and we get

\(13x^{2}-177x+602=0\)

Using the quadratic equation, we find two values of x. We get

\(x=7\\ x=\frac{86}{13}\)

Plugging these x values into the second equation where we isolated y, we get thatr \(y=7;y=\frac{66}{13}\)

So our final two answers are (7, 7) and (86/13, 66/13)

Thanks! :)

First, let's simplify the equation. 

Combining all like terms and reeaaranging the left, we get the equation

\(y=-5x^{2}-6x+10\)

Now, we know the x value of the vertex can be written in the form \(-\frac{b}{2a}\) for quadratics in the form \(y=ax^2+bx+c\)

Plugging in what we have, we get \(-\frac{6}{10}=-0.6\)

Subsituting this back to find y, we get \(y=11.8\)

Thus, our final answer is \((-0.6,11.8)\)

Thanks! :)

The answer is (-0.6, 11.8)

The answer is \(10 \cdot 4 \cdot 720 =28,800\)

The answer is \(a \equiv 5 \pmod{7}\), and we have

\(a + 1 \equiv 5+1 \pmod{7}\)

So 6 is our answer. 

Let's calculate the number of ways we can seat the children and adults seperately. 

First, there are \(4! = 24\) ways to seat the children and \(4! = 24 \) ways to seat the adults. 

We now multiply these values together to get

\(24*24=576\)

So 576 is our answer. 

Thanks! :)

Since we have \(a \equiv 5 \pmod{7}\), then we have

\(a + 1 \equiv 5+1 \pmod{7}\)

So 6 is the answer. 

There actually is no minimum value. 

The larger x gets, the smaller the fraction becomes, therefore meaning there is no minimum value. 

The range is \(0 <1\) , meaning no minimum value is present. 

Here's an image

The horizontal asymtote is \(y=0\)

Thanks! :)

Player 1 has a 1/6 (1 in 6) chance of winning because there are 6 numbers that he can roll but only one that he wants to roll.

The answer is 17n - 8.

(a) Let w = a + bi and z = c + di.  The rest is expanding.

(b) Let w = a + bi and z = c + di.  Then

\[\dfrac{w + z}{1 + wz} = \dfrac{a + c + bi + di}{1 + (a + bi)(c + di)}\]

To express this in rectangular form, we can multiply the numerator and denominator by the conjugate:

\[\dfrac{a + c + bi + di}{1 + (a + bi)(c + di)} = \dfrac{(a + c + bi + di)((1 - (a + bi)(c + di))}{(1 + (a + bi)(c + di))(1 - (a + bi)(c + di))}\]

The denominator simplifies to (1 - (a^2 + b^2)(c^2 + d^2)), which is real.  The numerator simplifies to a^2 - b^2 + c^2 - d^2, which is also real.  Therefore, the complex number (z + w)/(zw + 1) is real.

The answer is \(113\)

The answer is \(m=0; k =2\)

The answer is \(1+2+6=9\)

\(9-7=2\)

Since we have\(r = .036 / .4 = 36 / 400 = 9 /100 = .09 \), our answer is

\(.4 / [ 1 -.09] = .4 / .91 = 40 /91\)

the answer is \(2\)

The answers to this question are \(2,4\)

the answers are \(2,4\)

the remainder is \(.652173913*23\approx 15\)