Binomial Theorem and Logarithms: For the Use of the Midshipmen at the Naval School, PhiladelphiaPerkins & Purves, 1843 - 92 páginas |
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Página 71
... substitute for x and A their values formula ( 40 ) becomes 1 x = log.b , A == log . b b = 1 + 1 / log . 2 + M M 1 ( 108 , 0 ) 2 + 1 ( 108 , 6 ) 3 1 / log . 1 log.b 4 . 1 log . b 5 + 2.3.4.5 2.3 M 1 + + ( 108 ( log . b ) + & c . 2.3.4.5 ...
... substitute for x and A their values formula ( 40 ) becomes 1 x = log.b , A == log . b b = 1 + 1 / log . 2 + M M 1 ( 108 , 0 ) 2 + 1 ( 108 , 6 ) 3 1 / log . 1 log.b 4 . 1 log . b 5 + 2.3.4.5 2.3 M 1 + + ( 108 ( log . b ) + & c . 2.3.4.5 ...
Página 76
... substitute in ( 49 ) the value of M given by ( 50 ) , we have log . b = L. b L. a • ( 51 ) That is , the logarithm of a number to any given base is equal to the naperian logarithm of that number divided by the naperian logarithm of the ...
... substitute in ( 49 ) the value of M given by ( 50 ) , we have log . b = L. b L. a • ( 51 ) That is , the logarithm of a number to any given base is equal to the naperian logarithm of that number divided by the naperian logarithm of the ...
Página 79
... substitute n + d = p + 1 , d 2n = 2p + 1 , 2d = 1 , and n log . ( p + 1 ) = log.p + 2M ( ; 1 2p + 1 1 + 3n3 n 5n5 n - d = p , whence we have ; the formula becomes 1 M ( 2p + 1 +3 ( 2p + 1 ) + 5 ( 2p + 1 ) + & c . ) ( 64 ) ...
... substitute n + d = p + 1 , d 2n = 2p + 1 , 2d = 1 , and n log . ( p + 1 ) = log.p + 2M ( ; 1 2p + 1 1 + 3n3 n 5n5 n - d = p , whence we have ; the formula becomes 1 M ( 2p + 1 +3 ( 2p + 1 ) + 5 ( 2p + 1 ) + & c . ) ( 64 ) ...
Página 82
... substitute in ( 72 ) the value of M , given by ( 42 ) , we shall have log.p = 1 2 » [ p = —1— ¦ ( p == — 1 ) 2 + ¦ ( p = —1 ) ' — & c . ] n 1 m [ 1 3 2 1 am - 1- 1— ( a " — 1 ) 2 + ( a − 1 ) ' — & c . ] 3 ( 73 ) which is the formula ...
... substitute in ( 72 ) the value of M , given by ( 42 ) , we shall have log.p = 1 2 » [ p = —1— ¦ ( p == — 1 ) 2 + ¦ ( p = —1 ) ' — & c . ] n 1 m [ 1 3 2 1 am - 1- 1— ( a " — 1 ) 2 + ( a − 1 ) ' — & c . ] 3 ( 73 ) which is the formula ...
Página 84
... substitute numbers for letters , and compute the series by adding a number of terms , greater or less according to the degree of accuracy required . It will be observed that , generally , the most convergent formulæ are those which re ...
... substitute numbers for letters , and compute the series by adding a number of terms , greater or less according to the degree of accuracy required . It will be observed that , generally , the most convergent formulæ are those which re ...
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Binomial Theorem and Logarithms: For the Use of the Midshipmen at the Naval ... William Chauvenet Sin vista previa disponible - 2017 |
Términos y frases comunes
2n³ 3n³ 3d power 3d root 4th power 4th root Algebra anti-logarithms approximate ax=b binomial theorem Briggs calculation CHAPTER common logarithms compute convenient convergent cube root decimal fraction decimal point denominator example exponential equation express the value find log find the square find the value finite number formula becomes fractional exponents given logarithm given number Hence Hutton indefinitely small infinite series integral exponents involution and evolution log.b loga m+1)th term manner method modulus multiply naperian logarithm Newton number is equal number of terms obtain places of decimals positive integer power of a+b power or root powers and roots prime numbers quantity reciprocal rithms root of a³ significant figure square root succeeding terms system of logarithms system whose base uneven unit's place unity values substituted whence
Pasajes populares
Página 50 - The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.
Página 49 - The logarithm of the product of two or more numbers is equal to the sum of the logarithms of those numbers.
Página 61 - The fourth term is found by multiplying the second and third terms together and dividing by the first § 14O.
Página 50 - The logarithm of the quotient of two numbers is equal to the logarithm of the dividend minus the logarithm of the divisor.
Página 19 - Cxz+, etc.=A'+B'x+C'z2 + , etc., must be satisfied for each and every value given to x, then the coefficients of the like powers of x in the two members are equal each to each.
Página 74 - The logarithm of a number in any system is equal to the Naperian logarithm of that number multiplied by the modulus of the system.
Página 49 - Corollary. When the base is less than unity, it follows, from art. 3, that the logarithms of all numbers greater than unity are negative, while those of all numbers less than unity are positive. But when, as is almost always...
Página 55 - ... place, the characteristic being positive when this figure is to the left of the units' place, negative when it is to the right of the units' place, and zero when it is in the units
Página 27 - I have no doubt that he made the difcovery himfelf, without any light from Briggs, and that he thought it was new for all powers in general, as it was indeed for roots and quantities with fractional and irrational exponents.
Página 50 - Bee that to divide one number by another, we subtract the log. of the divisor from the log. of the dividend, and the remainder is the log.