Nature Edit ①;②;③ negative numbers and 0 have no logarithm. ④* = 1;Identity and ProofA^log (a) (n) =n (a>0, a≠1) Derivation: Log (a) (a^n) =n identity proof at a>0 and a≠1,n>0: When log (a) (n) =t, satisfy (t∈r) there is a^t=n;a^ (log (a) (n)) =a ^t=n the proof is complete.Algorithmic editing ①②③ (M,n∈r) If, then M is a natural logarithm of a, i.e., e=2.718281828 ... For the base of the natural logarithm. Definition: If the basic properties: 1, 2, 3, 4, 5, deduction: 1, because, substituting, that is. 2, MN=MXN by the basic properties of 1 (replace M and N) by the nature of the exponent and because the exponential function is a monotone function, so 3, and (2) similar to the processing M/n=m÷n by the basic properties 1 (replace M and N) by the nature of the exponent and because the exponential function is a monotone function, so 4, and (2) similar processing by Replace m) by the nature of the exponent and since the exponential function is a monotone function, so or by the Basic property 2 (unfold,)logarithmic basic Properties 4 derivation processBasic Properties 4 The generalization is deduced as follows: By the formula for the change of the bottom (see below) [Yes, E is called the natural logarithm of the bottom] the derivation of the formula: set it to be: by the basic properties 4 can be obtained by changing the bottom formulaChange Bottom Formula edit Set B=a^m,a=c^n, then b= (c^n) ^m=c^ (MN) ..... ....... ................ ① the logarithm of the base of a to ①, there is: Log (a) (b) =m...................................② to the ① to take the logarithm of the base C, there is: Log (c) (b) =mn ..... .... ....... ③③/②: Log (c) (b)/log (a) (b) =n=log (c) (a) ∴log (a) (b) =log (c) (b)/log (c) (a) Note: log (a) (b) indicates a logarithm of the base x. Expansion of the formula for changing the bottom: Based on the base of E and a formula substitution: logae=1/(LNA)Derivation Formula edit Log (1/a) (1/b) =log (a^-1) (b^-1) =-1/-1logab=loga (b) Loga (b) *logb (a) =1derivative number edit (Xlogax) ' =logax+1/lna where a is the base of Logax, and X is the true number; (Logax) ' =1/xlna special a=e (logex) ' = (LNX) ' =1/x
Calculation formula for logarithm
