Logarithm Equations
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Questions and Answers

What is the equivalent of log(x) = 3?

  • x^3 = log(e)
  • ln(x) = 3 (correct)
  • e^3 = x
  • 10^3 = x

    • What is the base of natural logarithms?

  • ln(e)
  • 10
  • e (correct)
  • 2

    • In the given triangle ABC, what is true about the sides?

  • a = b
  • b = 2a
  • a = 2b (correct)
  • a > b

    • What can be said about angle C in triangle ABC?

    <p>cos(C) = -1/2</p> Signup and view all the answers

    What is true about the circle with center O?

    <p>Segment AB is always a secant</p> Signup and view all the answers

    What is true about quadrilateral ABCD?

    <p>∠𝐵 = ∠𝐷</p> Signup and view all the answers

    What are the coordinates of the other endpoint?

    <p>(4, 5)</p> Signup and view all the answers

    What is the equation of the line through (1,-1)?

    <p>x + 2y = 1</p> Signup and view all the answers

    What is the name of a polygon with 40 sides?

    <p>tetracontagon</p> Signup and view all the answers

    What is the value of p if the area of a rectangle decreases by 12% when its length is increased by 10% and its width is decreased by p%?

    <p>20</p> Signup and view all the answers

    What is the name of a straight line segment whose endpoints lie on a circle?

    <p>chord</p> Signup and view all the answers

    What is the solution to the equation Arctan x + Arctan (1 - x) = Arctan 4/3?

    <p>x = 1/2</p> Signup and view all the answers

    What is the speed of the boat?

    <p>427.88 ft/min</p> Signup and view all the answers

    What is the general form of the equation of a circle with center at (3,1) and tangent to the line 4x + 3y + 5 = 0?

    <p>x + y - 6x - 2y - 6 = 0</p> Signup and view all the answers

    What is the shift of the graph 𝑦 = sin (𝑥 + )?

    <p>left 1 unit</p> Signup and view all the answers

    What is the base edge of a regular triangular pyramid with an altitude of 9 m and a volume of 187.06 cu.m?

    <p>6</p> Signup and view all the answers

    What is the approximate number of votes received by the first candidate in a city council election?

    <p>10,656</p> Signup and view all the answers

    What is the equation of the perpendicular lines?

    <p>y = 3x + 2 and y = 3x − 2</p> Signup and view all the answers

    What is the range of 𝑓(𝑥) = 2𝑥 − 6𝑥 + 7?

    <p>𝑓(𝑥) ≥ 7</p> Signup and view all the answers

    What is the simplified form of the expression +?

    <p>3 cos A</p> Signup and view all the answers

    What is the equation of the ellipse?

    <p>2𝑥 − 17𝑥 + 9 = 0</p> Signup and view all the answers

    What is the height of the triangle?

    <p>(𝑥 − 2)</p> Signup and view all the answers

    What is the volume of the cube?

    <p>67.43 cm3</p> Signup and view all the answers

    What is the range of the radio equipment?

    <p>±3𝑖</p> Signup and view all the answers

    What is the ratio of the volume of the sphere to the volume of the cylinder if a sphere is inscribed in a cylinder?

    <p>one-half</p> Signup and view all the answers

    Can two ships, one 155 km N42 40′E and the other 165 km N45 10′W of a shore station, communicate directly?

    <p>No, they are 215.0347 km apart</p> Signup and view all the answers

    What is the volume of water in a spherical tank having a diameter of 4 m?

    <p>5.236 m3</p> Signup and view all the answers

    What is the depth of the water in a spherical tank having a diameter of 4 m and a volume of 5.236 m3?

    <p>1.0 m</p> Signup and view all the answers

    What is the volume of a regular triangular prism with a base edge of 2 m circumscribing a sphere of radius 3 m?

    <p>9.59 m3</p> Signup and view all the answers

    A right circular cone is cut into two parts of equal weight. Where should the cut be made?

    <p>0.595h</p> Signup and view all the answers

    What is the volume of the portion of a sphere included between two parallel circular sections with radii 4 in. and 4.8 in.?

    <p>The volume can be calculated using the radii of the sections.</p> Signup and view all the answers

    What is the total surface area of a right circular cone?

    <p>πr(r + l)</p> Signup and view all the answers

    What is the volume of the right circular cone with base radius 10 cm and altitude 15 cm?

    <p>314.4 in3</p> Signup and view all the answers

    What is the lateral surface area of a closed cylindrical tank with height 10 ft and diameter 4 ft?

    <p>384.52 cm2</p> Signup and view all the answers

    How many iron rivets can be made from two kilograms of metal?

    <p>Cannot be determined</p> Signup and view all the answers

    What is the height of the water in a horizontal cylindrical tank?

    <p>3.5 ft</p> Signup and view all the answers

    What is the height of the water in a hexagonal prism-shaped flower vase?

    <p>34.88 in</p> Signup and view all the answers

    What is the lateral area of a right section of an oblique prism with a hexagonal base?

    <p>428 cm3</p> Signup and view all the answers

    What is the value of h'(x) given h(x) =

    <p>h'(x) = x + 1</p> Signup and view all the answers

    What is the value of p for which (x - p)(x + 3 - p) has a remainder of 4?

    <p>5, 1</p> Signup and view all the answers

    Study Notes

    Equations and Logarithms

    • Solving the equation log(x) = 3 is equivalent to solving ln(x) = 3, but with different numerical answers.
    • ln(x) is equivalent to loge(x), where e is an irrational number approximately equal to 2.71828.
    • Common logarithms (log) are typically used with a base of 10, while natural logarithms always use base e.

    Geometry

    • A chord is an extension of a chord in a circle which is a straight line segment of which the endpoints lie on the circle.
    • A polygon with 40 sides is called a tetracontagon.
    • In a quadrilateral ABCD, if AB = CD and ∠A = ∠C, then ∠B = ∠D.

    Trigonometry

    • sin(A) + sin(1 - A) = sin(4/3)
    • Simplify + as 3cos(A) or 2sec(A)tan(A)

    Analytic Geometry

    • The general form of the equation of a circle with center at (3,1) and tangent to the line 4x+3y+5=0 is x + y - 6x - 2y - 6 = 0.
    • The equation of an ellipse satisfying certain conditions is + =1 or + =1.

    Three-Dimensional Geometry

    • The volume of a sphere is that of a cylinder, and one-half that of a cylinder if the sphere is inscribed in the cylinder.
    • If a sphere is inscribed in a cylinder, the volume of the sphere is one-half that of the cylinder.

    Miscellaneous

    • In a right triangle, the length of the leg opposite the 45-degree angle is the geometric mean of the hypotenuse and the leg adjacent to the 45-degree angle.
    • If two ships have radio equipment with a range of 200 km, they can communicate directly if they are within 190.0234 km of each other.

    I hope this helps! Let me know if you need further clarification.

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    Description

    Compare the solutions of logarithmic equations, including log(x) = 3 and ln(x) = 3, and identify the properties of their solution sets.

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