Log x + log (x - 6) =3

x = 3 + sqrt(e^3 + 9)

Step-by-step explanation:

Solve for x over the real numbers:

log(x) + log(x - 6) = 3

Hint: | Combine logarithms.

log(x - 6) + log(x) = log(x (x - 6)):

log(x (x - 6)) = 3

Hint: | Eliminate the logarithm from the left hand side.

Cancel logarithms by taking exp of both sides:

x (x - 6) = e^3

Hint: | Write the quadratic polynomial on the left hand side in standard form.

Expand out terms of the left hand side:

x^2 - 6 x = e^3

Hint: | Take one half of the coefficient of x and square it, then add it to both sides.

Add 9 to both sides:

x^2 - 6 x + 9 = e^3 + 9

Hint: | Factor the left hand side.

Write the left hand side as a square:

(x - 3)^2 = e^3 + 9

Hint: | Eliminate the exponent on the left hand side.

Take the square root of both sides:

x - 3 = sqrt(e^3 + 9) or x - 3 = -sqrt(e^3 + 9)

Hint: | Look at the first equation: Solve for x.

Add 3 to both sides:

x = 3 + sqrt(e^3 + 9) or x - 3 = -sqrt(e^3 + 9)

Hint: | Look at the second equation: Solve for x.

Add 3 to both sides:

x = 3 + sqrt(e^3 + 9) or x = 3 - sqrt(e^3 + 9)

Hint: | Now test that these solutions are correct by substituting into the original equation.

Check the solution x = 3 - sqrt(e^3 + 9).

log(x) + log(x - 6) ⇒ log((3 - sqrt(e^3 + 9)) - 6) + log(3 - sqrt(e^3 + 9)) = 2 i π + 3 ≈ 3 + 6.28319 i:

So this solution is incorrect

Hint: | Check the solution x = 3 + sqrt(e^3 + 9).

log(x) + log(x - 6) ⇒ log(3 + sqrt(e^3 + 9)) + log((3 + sqrt(e^3 + 9)) - 6) = 3:

So this solution is correct

Hint: | Gather any correct solutions.

The solution is:

x = 3 + sqrt(e^3 + 9)

9514 1404 393

  x = 3 + √1009 ≈ 34.765

Step-by-step explanation:

Take the antilog and solve the resulting quadratic.

  x(x -6) = 10^3

  x^2 -6x +9 = 1009 . . . . . . add (-6/2)^2 to complete the square

  (x -3)^2 = 1009 . . . . . . . . . write as a square

  x = 3 + √1009 ≈ 34.765 . . . . . square root and add 3

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Only the positive root makes any sense, since the log of a negative number is not defined (in real numbers).

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For the graph, we subtracted 3 so we are looking for the x-intercept where f(x)=0.

pentagon

step-by-step explanation:

the coordinates   a(5, -2), b(3,-1),c(-4,-4) d(-3,8) and e(-1,4) forms the vertices of the polygon when they are connected in order.

in order to classify that which polygon is it with the given coordinates, first count the number of vertices and then count the number of sides it can make with these vertices.

since, the given coordinates have five vertices, therefore, it will make five sides. moreover, pentagon is regular in nature therefore it will be convex.

now, a polygon with five sides is known as pentagon, therefore, the given polygon is the pentagon.