The basis of logarithmic function is fraction.

The power of the base can be raised as the coefficient in front of the whole logarithmic function, but you should pay attention to the fact that the power of the base is raised as the denominator, that is, you regard the high-order coefficient as a fraction, the power of the base is advanced as the denominator, and then the power of the real number is advanced as the numerator.

Expand:

logarithmic function

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 aN=b, where a is called the base of logarithm and n is called a real number.

Axiomatic definition of logarithmic function

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 zero.

Cardinality is greater than 0 instead of 1.

Why is the base of logarithmic function greater than 0 instead of 1?

In the ordinary logarithmic formula, a

The general form of logarithmic function is y=log(a)x, which is actually the inverse function of exponential function and can be expressed as x = a y, so the stipulation of a in exponential function is also applicable to logarithmic function.

The figure on the right shows the function diagram of different size 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.

The domain of (1) logarithmic function is a set of real numbers greater than 0.

(2) The range of logarithmic function is the set of all 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 abbreviations for 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, then:

( 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)

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

The natural logarithm of Ln is based on e, and e is an infinite acyclic decimal.

Lg's common logarithm is based on 10 [edit 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 logarithm with the base of N, and it is recorded as log(a)(N)=b, where A is called the base of logarithm and N is called real number.

Cardinality is greater than 0 instead of 1.

Operational properties of logarithm:

When a>0 and a≠ 1, m >；; 0, N>0, then:

( 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)

(4) the formula of bottoming: log (a) m = log (b) m/log (b) a (b >); 0 and b≠ 1)

Relationship between logarithm and exponent

When a>0 and a≠ 1, a x = n x = ㏒ (a) n (logarithmic identity).

Common abbreviations for logarithmic functions:

(1) common logarithm: lg(b)=log( 10)(b)

(2) natural logarithm: ln(b)=log(e)(b)

E=2.7 1828 1828 ... usually only the definition of logarithmic function is adopted.

The general form of logarithmic function is y=㏒(a)x, which is actually the inverse function of exponential function (Y = x = a y inverse function of two functions symmetrical about a straight line), and can be expressed as x = A Y. Therefore, the adjustment of A in exponential function (a >；); 0 and a≠ 1) are also applicable to logarithmic functions.

The figure on the right shows the function diagram of different size 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 this paragraph] Attribute domain: (0, +∞) Value domain: 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: the bottom is true and the logarithm is positive.

True heteronegativity at the bottom