The concept of 1 logarithm if a (a >; 0, and the b power of a≠ 1) is equal to n, that is, ab=N, then the number b is called the logarithm of n with a base, and is recorded as: logaN=b, where a is called the base of logarithm and n is called a real number. According to the definition: ① Negative numbers and zero have no logarithm; ②a & gt； 0 and a ≠ 1, n >；; 0; (3) loga1= 0, logaa = 1, alogan = n, logaab = B. Especially the logarithm with the base of 10 is called ordinary logarithm, which is called log 10N and abbreviated as lgN；; The logarithm with the irrational number e (e = 2.718 28 ...) as the base is called natural logarithm, which is recorded as logeN and abbreviated as lnN.2 The reciprocal formula of logarithm and exponent is called abN exponent ab=N (base) (exponent) (exponent) logaN=b (base) (logarithm) (real number). 0, a≠ 1, M>0, N>0, then (1) loga (Mn) = logam+Logan. ②loga Mn = logam-Logan。 (3) logamn = nlogam (n ∈ r)。 Q: ① why should condition a > be added to the formula? 0，a≠ 1，M & gt0，N & gt0? ②logaan=？ (n∈R) ③ Comparison between logarithmic formula and exponential formula. (Students fill in the form) Formula ab=NlogaN=b Name A- Power Base B-N-A- Logarithmic Base B-N- Operation Quality AM An = AM+N AM÷An =(AM)N =(A > 0 and a ≠ 1, N ∈ R. A≠ 1, M>0, N>0) In the definition of logarithm, why should a > 0, a≠ 1 be specified? The reasons are as follows: ① If a < 0, some values of n do not exist, such as log-28? ② If a=0, then b does not exist when N≠0; When N=0, b is not unique. Can it be any positive number? ③ If a= 1, then b does not exist when N≠ 1; When N= 1, b is not unique and can be any positive number? In order to avoid the above situation, it is stipulated that the base of logarithmic formula is a positive number that is not equal to 1 Problem solving skills 1 (1) Write the following exponential formula as a logarithmic formula: ① 54 = 625; ②2-6= 164; ③3x = 27； ④ 13m=5？ 73.(2) Write the following logarithmic formula as an exponential formula: ① log1216 =-4; ②log 2 128 = 7； ③log 327 = x； ④LG 0.0 1 =-2； ⑤ln 10 = 2.303； ⑥ LG π = K. Analysis is defined by logarithm: ab=N? Logan = B. Solution (1)1log5625 = 4.2log2164 =-6.3log327 = x.4log135.73 = m. The solution to the problem is to firmly grasp the definition of logarithm: ab = ② 27 = 128.③ 3x = 27。 ④ 10-2 = 0.0 1.5e 2.303 = 65438+。 (2)log2(log5x)= 0； (3)logx 27 = 3 1+log32； (4)logx(2+3)=- 1。 Analyze the (1) logarithmic exponential formula and get: x=8-23=? (2)log5x=20= 1.x=？ (3)3 1+log32=3×3log32=？ 27=x？ (4)2+3=x- 1= 1x.x=？ Answer (1) x = 8-23 = (23)-23 = 2-2 =14. (2) log5x = 20 = 1，x = 5 1 = 5。 (3) logx27 = ∴x= 12+3=2-3. Problem solving skills ① The idea of reduction is an important mathematical thought, and the logarithmic formula is closely related to the exponential formula. When solving related problems, the two forms often transform each other. ② Clever use of formulas: loga 1 = 0, logaa = 1. Logay=5, and find the value of a = [x 3x- 1y2] 12. Analytic thinking 1: Given the value of logarithmic formula, we need exponential value, so we can convert logarithmic formula into exponential formula and then evaluate it by exponential operation; Idea 2: Take the logarithm of both sides of the exponential formula with the same base, and then evaluate it through the operation of the logarithmic formula? Solution1:logax = 4, logay = 5, ∴ x = a4, y = a5, ∴ a = x 512y-13 = (a4) 512 (a5