7th Grade Math 7th Grade Math Logarithms :- The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 10 × 10 × 10 = 103. More generally, if x = by, then y is the logarithm of x to base b, and is written y = logb(x), so log10(1000) = 3. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. They were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. Tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition because of the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors: The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century.
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The logarithm to base b = 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the constant e (≈ 2.718) as its base; its use is widespread in pure mathematics, especially calculus. The binary logarithm uses base b = 2 and is prominent in computer science. Geometric Solids :- solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. It was developed following the development of plane geometry. Stereometry deals with the measurements of volumes of various solid figures including cylinder, circular cone, truncated cone, sphere, and prisms. The Pythagoreans had dealt with the regular solids, but the pyramid, prism, cone and cylinder were not studied until the Platonists. Eudoxus established their measurement, proving the pyramid and cone to have one-third the volume of a prism and cylinder on the same base and of the same height, and was probably the discoverer of a proof that the volume of a sphere is proportional to the cube of its radius. Pyramids :- A pyramid (from Greek: πυραμίς pyramis) is a structure whose shape is roughly that of a pyramid in the geometric sense; that is, its outer surfaces are triangular and converge to a single point at the top. The base of a pyramid can be trilateral, quadrilateral, or any polygon shape, meaning that a pyramid has at least three outer triangular surfaces (at least four faces including the base). The square pyramid, with square base and four triangular outer surfaces, is a common version.
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A pyramid's design, with the majority of the weight closer to the ground,[2] and with the pyramidion on top means that less material higher up on the pyramid will be pushing down from above. This distribution of weight allowed early civilizations to create stable monumental structures. Pyramids have been built by civilizations in many parts of the world. For thousands of years, the largest structures on Earth were pyramids—first the Red Pyramid in the Dashur Necropolis and then the Great Pyramid of Khufu, both of Egypt, the latter the only one of the Seven Wonders of the Ancient World still remaining. Khufu's Pyramid is built entirely of limestone, and is considered an architectural masterpiece. Analytic geometry :- Analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and uses deductive reasoning based on axioms and theorems to derive truth. Analytic geometry is widely used in physics and engineering, and is the foundation of most modern fields of geometry, including algebraic, differential, discrete, and computational geometry. Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane (2 dimensions) and Euclidean space (3 dimensions).
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