LOGARITHMS. § 1. The logarithm of a number N is the exponent denoting the power to which a fixed number called the base must be raised in order to produce N. Thus, if N= b2, then is log, N= x; or, in words, when the base is b the logarithm of N is x. What is log28? What is log 16 16 ( 4 ...log+16 = 4. What is the number whose logarithm is 5 when the base is 2? log, N = 5; N=25=32. What is the number whose logarithm is 3 when the base is 3? Ans. 27. What is the number whose logarithm is 3 when the base is 10? Ans. 1000. What is the number whose logarithm is 2 when the base is 8? What is the number whose logarithm is base is 10? What is the number whose logarithm is base is 27? § 2. The logarithms of all numbers referred to the same base are said to belong to the same system of logarithms. Thus, log 106, log 107, log 1084, log 10 768, all belong to the denary or common system. § 3. In any system of logarithms the logarithm of the base itself is 1. § 4. In every system of logarithms the logarithm of 1 is 0. For 1 = 20; 1 = = ... log21 = 0: 10°; .. log 101 = 0: and, in general, 16°; ... log,1 = 0. § 5. In any system the logarithm of the reciprocal of a number is the negative of the logarithm of the number. What is the number whose logarithm is base is 16? N = 16 What is the number whose logarithm is base is 10? Ans. 3. 3 when the Ans. 0.001. § 6. In any system the logarithm of the product of two or more numbers is equal to the sum of the logarithms of the numbers. Proof: If then is now m = l = b2, ...log(lx m × n) = x + y + z = logl+ log m+logn. It is evident that this principle may be applied to any number of factors. What is log(8 x 4 x 32)? 82; ... log,8 = 3: 4 = 22; log 24 = 2: 3225; ... log,32 = 5: log2(8 x 4 x 32) = log,8 + log, 4+ log,32 =3+2+5=10. What is log.x3x81)? Ans. 2+1+4= 3. If log 10 123 = 2.0899, what is log 10 12300? 100 × 123 = 102 × 102.0899; ... log 10 12300 = 2 +2.0899 = 4.0899. 12300 If log 10 2 = 0.3010, what is log 10 200? Ans. 2.3010. § 7. In any system the logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator. Proof: If l=b", mb", then is log,l=x, log,m=y; Given: log 103 = 0.4771 and log 102 = 0.3010; what is 3 ? log 102 Ans. 0.1761. Given: log 10 123 = 2.0899; what is log 100.123? § 8. In any system the logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power. Proof: If l = b2, then is log,l=x, lm = (bx)m = bmx ; log 7m =mx=mx logl. Under this head may be brought a root, for a root may be regarded as a fractional power. |