If the real number formula has no root number, then as long as the real number formula is greater than zero, if there is a root number, the real number is required to be greater than zero, and the formula in the root number is greater than or equal to zero. Cardinality should be greater than 0 instead of 1? Why is the base of logarithmic function greater than 0 instead of 1? ? In an ordinary logarithmic formula? A<0, or = 1? When it is, there will be a corresponding b value. However, according to the logarithmic definition:? logaa = 1； If a= 1 or =0, then logaa can be equal to all real numbers (such as log 1? 1 can also be equal to 2, 3, 4, 5, etc. Second, according to the definition, the operation formula: loga? M^n？ =? nloga？ m？ If a

Edit the definition and operational properties of the logarithm in this paragraph.

Generally speaking, if the power of a (a is greater than 0, and a is not equal to 1) is equal to n, then this number b is called the logarithm of n with the base of a, and it is recorded as log(a)(N)=b, where a is called the base of logarithm and n is called a real number. ? Cardinality should be greater than 0 instead of 1? The real number is greater than 0.

Operational properties of logarithm:

When a>0 and a≠ 1, m >；; 0, N>0, so:? ( 1)log(a)(MN)= log(a)(M)+log(a)(N)； ? (2)log(a)(M/N)= log(a)(M)-log(a)(N)； ? (3)log(a)(M^n)=nlog(a)(M)？ (n∈R)？ (4) the formula for changing the bottom: log(A)M=log(b)M/log(b)A? (b>0 and b≠ 1)? (5)? a^(log(b)n)=n^(log(b)a)？ Proof: let a = n x? Then a (log (b) n) = (n x) log (b) n = n (x log (b) n) = n log (b) (n x) = n (log (b) a)? (6) Logarithmic identity: a log (a) n = n; ? log(a)a^b=b

Relationship between logarithm and exponent

When a>0 and a≠ 1, a x = n? x=㏒(a)N

Edit the logarithmic function of this paragraph.

The picture on the right shows the function diagram of different sizes A:? You can see that the graphs of logarithmic functions are only symmetric graphs of exponential functions about the straight line y=x, because they are reciprocal functions. ? ( 1)? The domain of logarithmic function is a set of real numbers greater than 0. ? (2)? The range of logarithmic functions is a set of real numbers. ? (3)? The function image always passes through the (1, 0) point. ? (4)? When a is greater than 1, it is monotone increasing function and convex; When a is less than 1 and greater than 0, the function is monotonically decreasing and concave. ? (5)? Obviously the logarithmic function is unbounded. ? Common abbreviated expressions of logarithmic functions:? ( 1)log(a)(b)=log(a)(b)？ (2)lg(b)=log( 10)(b)？ (3)ln(b)=log(e)(b)？ Operational properties of logarithmic function:? If a > 0 and a is not equal to 1, m >；; 0, N>0, so:? ( 1)log(a)(MN)= log(a)(M)+log(a)(N)； ? (2)log(a)(M/N)= log(a)(M)-log(a)(N)； ? (3)log(a)(M^n)=nlog(a)(M)？ (n belongs to r)? (4)log(a^k)(M^n)=(n/k)log(a)(M)？ (n belongs to r)? What is the relationship between logarithm and exponent? When a is greater than 0 and a is not equal to 1, the x power of a =N is equivalent to log(a)N? log(a^k)(M^n)=(n/k)log(a)(M)？ (n belongs to r)? Bottom-changing formula? (very important)? log(a)(N)=log(b)(N)/log(b)(a)=？ lnN/lna=lgN/lga？ Where is it? Natural logarithm? E-based？ E is an infinite acyclic decimal? lg？ Commonly used logarithm? Based on 10

Edit the common short expressions of logarithmic functions in this paragraph.

(1) common logarithm: lg(b)=log( 10)(b)? (2) natural logarithm: ln(b)=log(e)(b)? e=2.7 1828 1828...？ Usually we only take e=2.7 1828? What is the definition of logarithmic function? What is the general form of logarithmic function? Y=㏒(a)x is actually the inverse function of the exponential function (the images of the two functions are symmetrical about the straight line Y = X = a Y), which can be expressed as X = A Y. Therefore, the adjustment of A in the exponential function (a >；; 0 and a≠ 1) are also applicable to logarithmic functions. ? The picture on the right shows the function diagram of different sizes A:? You can see that the graphs of logarithmic functions are only symmetric graphs of exponential functions about the straight line y=x, because they are reciprocal functions.

Edit the properties of this paragraph.

Domain solution: logarithmic function y=loga? x？ The domain of is {x? | x & gt0}, but when solving the domain of logarithmic compound function, we should pay attention not only to the fact that the real number is greater than 0, but also to the fact that the radix is greater than 0 and not equal to 1. In order to solve the domain of the function y=logx(2x- 1), {x >； must be satisfied; 0 and x ≠ 1}? . ? {2x- 1 >0? =〉x & gt； 1/2 and x≠ 1, that is, what is its domain? {x？ | x> 1/2 and x ≠ 1} Range: real number set r? Fixed point: The function image always passes through the fixed point (1, 0). ? Monotonicity: when a> is 1, it is monotone increasing function and convex on the domain; ?

When 0<a< 1, it is a monotonic decreasing function in the definition domain and is concave. ? Parity: Non-odd and non-even functions, or no parity. ? Periodicity: Not a periodic function? Zero: x= 1? Note: Negative numbers and 0 have no logarithm. ? Two classic words: Is the bottom true logarithm positive? True heteronegativity at the bottom

Edit the history of this logarithmic function:

From the end of 16 to the beginning of 17, at that time, the development of natural sciences (especially astronomy) often encountered a large number of accurate and huge numerical calculations, so mathematicians invented logarithms in order to seek simplified calculation methods. ? Two series of integer arithmetic written by German Steven (1487- 1567) with 1544. On the left is the geometric series (called the original number), and on the right is arithmetic progression (called the representative of the original number, or called the Exponent, and German is Exponent? , which means representative). ? If you want to find the product (quotient) of any two numbers on the left, you only need to find the sum (difference) of its representative (exponent) first, and then put this sum (difference) on a primitive number on the left, then this primitive number is the product (quotient) you want. Unfortunately, Steve did not explore further and did not introduce the concept of logarithm. ? Napier is quite good at numerical calculation. The "Napier algorithm" he created simplifies the multiplication and division operation, and its principle is to replace multiplication and division with addition and subtraction. ? His motivation for inventing logarithm is to find a simple method to calculate spherics. He constructed the so-called logarithmic square based on a very unique idea related to particle motion. Method, the core idea of which is the connection between arithmetic progression and geometric sequence. He expounded the principle of logarithm in the description of wonderful logarithm table, which was later called? Napier logarithm, written as nap. X, what is its relationship with natural logarithm? Take a nap. ㏒x= 107㏑( 107/x)？ Therefore, Napier logarithm is neither a natural logarithm nor an ordinary logarithm, which is far from today's logarithm. ? The Swiss piccard (1552- 1632) also independently discovered logarithms, probably earlier than Napier, but published later (1620). ? Briggs of Britain created the ordinary logarithm in 1624. ? 16 19, the new logarithm written by Peter in London makes the logarithm closer to the natural logarithm (based on e=2.7 1828 ...). The invention of logarithm played an important role in the development of society at that time. As the scientist Galileo (1564- 1642) said, "Give me time, space and logarithm, and I can create a universe". ? Another example is18th century mathematician Laplace (? 1749- 1827) also mentioned: "Logarithmically shortening the calculation time doubles the life span of astronomers". ? Proportion and Logarithm, the first logarithmic work introduced to China, was compiled by Polish Muniz (161-kloc-0/-656) and China Xue Fengzuo in the middle of17th century? Made it up. At that time, in lg2=0.30 10, 2 was called "real number", 0.30 10 was called "pseudo number", and the real number and pseudo sequence were in one table, so it was called logarithmic table. Then you changed careers? "Pseudo number" is "logarithm". ? Dai Xu (1805- 1860), a mathematician in the Qing Dynasty, developed a variety of quick methods for finding logarithm, including logarithmic simplification (1845) and continuous logarithmic simplification (1846). 1854, British mathematician Yue Se (1825- 1905)? I was very impressed after seeing these works. ? Nowadays, middle school mathematics textbooks all talk about "exponent" first, and then introduce the concept of "logarithm" in the form of inverse function. But in history, on the contrary, the concept of logarithm did not come from index, because there was no clear concept of fractional index and irrational index at that time. Briggs once suggested to Napier that logarithm should be expressed by power exponent. 1742? J William (1675- 1749) wrote a preface for G William's logarithm table, in which the exponent can define logarithm. And Euler in his masterpiece infinitesimal? Analysis and discussion (1748) clearly puts forward that logarithmic function is the inverse function of exponential function, which is consistent with the current textbooks.