Mathematical logarithmic function properties, what are logarithmic function properties?

Basic Properties:

1、a^(log(a)(b))=b

2、log(a)(a^b)=b

3、log(a)(mn)=log(a)(m)+log(a)(n);

4、log(a)(m÷n)=log(a)(m)-log(a)(n);

5. log(a)(m n)=nlog(a)(m)6、log(a n)m=1 nlog(a)(m) Other properties: 1Bottom change formula.

log(a)(n)=log(b)(n)÷log(b)(a)3.The image of the logarithmic function is past (1,0) points.

4.For the y=log(a)(n) function, when 01, the function is (0,+ single increase), and as a increases, the image gradually rotates counterclockwise with (point as the axis, but not more than x=1.

5.As with the image relationship between other functions and inverse functions, the image of the logarithmic function and the exponential function is symmetrical with respect to the straight line y=x.

1。The images are all on the right side of the y-axis, and the definition domain is 0 to positive infinity.

2。Function images are all past (1,0) point 1 is a logarithm of zero.

3。Looking from left to right, when a>1, the image gradually rises, when 0,1, y=log a x (**Du Niang does not let you send ah) is the increase function When 0 a 1, y=log a x is the subtraction function.

4。When a 1, the ordinate of the function image to the right of point (1,0) is greater than 0, and the ordinate to the left of point (1,0) is less than 0When 0 a 1, the image is the opposite, the ordinate to the right of point (1,0) is less than 0, and the ordinate to the left of point (1,0) is greater than 0

When a 1 x 1, then y=log a x 00 x 1, y=log a x 0

When 0 a 1 x 1, then y=log a x 0 0 x 1, y=log a x 0

The logarithmic function properties are as follows:1. Range: the set of real numbers r, obviously the logarithmic function is unbounded;

2. Fixed point: the image of Hui Chenchen function is constant over the fixed point (1,0);

3. Monotonicity.

a>1, in the definition of the domain.

The upper is a monotonic increasing function;

4. Parity: non-odd and non-even functions;

5. Periodicity: not a periodic function;

6. Zero point: x=1;

7. The base number should be "0" and ≠1 true number" 0, and when comparing the values of two functions: if the base numbers are the same, the larger the true number, the larger the function value. (a>1); If the bases are the same, the smaller the true number, the larger the function value (0<>

Logarithmic function expression:1) Common logarithm: lg(b)=log10b (10 is the base).

2) Natural logarithms.

ln(b)=logeb (e is the base).

e is an infinite non-cyclic decimal pre-Zen.

Normally, only e= is taken.

The graph of the logarithmic function is nothing but an exponential function.

The symmetrical graphs of the graph with respect to the straight line y=x, because they are inverse functions of each other.

In general, logarithmic functions have powers (true numbers) as independent variables, exponents as dependent variables, and bases as constants.

The logarithmic function is one of the 6 basic elementary functions. where the definition of logarithm:

If ax=n(a>0, and a≠1), then the number x is called the logarithm of base n with a, denoted as x=logan, read as the logarithm of base n with a, where a is called the base of the logarithm and n is called the true number.

In general, the function y=logax(a>0, and a≠1) is called a logarithmic function, that is, a function with power (true number) as the independent variable, exponent as the dependent variable, and base number as the constant, which is called the logarithmic function.

where x is the independent variable and the domain of the function is (0, + i.e., x>0. It is actually the inverse of the exponential function, which can be expressed as x=ay. Therefore, the requirement for a in exponential functions also applies to logarithmic functions.

"log" is an abbreviation of the Latin logarithm (logarithm), which reads: [English] [l ɡ] American [l ɡ, lɑɡ].

The properties of the logarithmic function are:

A logarithmic function uses power (true number) as an independent variable.

The index is the dependent variable.

The base is a function of a constant. Logarithmic functions are 6 classes of basic elementary functions.

One. The definition of logarithm: if ax=n(a>0, and a≠1), then the number x is called the logarithm of a base n, denoted as x=logan, and read as the logarithm of a base n, where a is called the base of the logarithm and n is called the true number.

In general, the function y=logax(a>0, and a≠1) is called a logarithmic function, that is, a function with power (true number) as the independent variable, exponent as the dependent variable, and base number as the constant, which is called the logarithmic function.

Logarithmic functions withPrime numbersFunctions:

The general form of a logarithmic function is y= ax, which is essentially an exponential function.

The inverse function of (the image of two functions symmetrical with respect to the line y=x is the inverse of each other), which can be expressed as x=ay.

Therefore, for the exponential function for a (a>0 and a≠1), the figure on the right gives a graph of the function represented by different magnitudes a: with respect to x-axis symmetry.

When a>1, the larger a is, the closer the image is to the x-axis, and when 0 can be seen, the graph of the logarithmic function is nothing more than a symmetrical graph of the graph of the exponential function with respect to the line y=x, because they are inverse functions of each other.

Properties of logarithmic functions:In general, the function y=logax(a>0, and a≠1) is called a logarithmic function, that is, a function with power (true number) as the independent variable, exponent as the dependent variable, and base number as the constant, which is called the logarithmic function.

where x is the independent variable and the domain of the function is (0, + i.e., x>0. It is actually the inverse of the exponential function, which can be expressed as x=ay. Therefore, the requirement for a in exponential functions also applies to logarithmic functions.

Generated History:

In the late 16th and early 17th centuries, mathematicians invented logarithms in search of simplified calculations in the development of natural sciences (especially astronomy).

In his 1544 book "Arithmetic of Integer Numbers", Steffield (1487-1567) of Germany wrote two series of numbers, the left is a series of proportional numbers (called the original number), and the right is a series of equal difference numbers (called the representative of the original number, or exponent, the German word is exponent, which means representative).

Definition domain solving: The definition domain of the logarithmic function y=logax is, but if you encounter the solution of the definition domain of the logarithmic composite function, in addition to paying attention to greater than 0, you should also pay attention to the base greater than 0 and not equal to 1, for example, to find the definition domain of the function y=logx(2x-1), you need to meet both x>0 and x≠1

and 2x-1>0 to get x>1 2 and x≠1, i.e., the domain of its definition is the value range: the set of real numbers r, which is obviously unbounded by the logarithmic function;

fixed-point: the function image of the logarithmic function is constant with a fixed-point (1,0);

Monotonicity: a>1, it is a monotonic increasing function on the defined domain;

0 Parity: Non-odd and non-even functions.

Periodicity: Not a periodic function.

Symmetry: None.

Maximum: None. Zero point: x=1

Note: There is no logarithm for negative and 0.

Two classic sayings: the bottom truth is the same logarithmic coarse family positive, and the bottom truth is different and the logarithm is negative. The explanation is as follows:

That is, if y=logab (where a>0,a≠1,b>0) <>

When 00; When a>1, b>1, y=logab>0;

When 01, y=logab<0;

When a>1, 0 operates on properties.

In general, if a(a>0 and the power of b of a≠1) is equal to n, then the number b is called the logarithm of n with a as the base, denoted as logan=b, where a is called the base of the logarithm and n is called the true number.

The base number should be 0 and ≠1 true number" 0

And, when comparing two function values:

If the bases are the same, the larger the true number, the larger the function value. (a>1) If the bottom stool number is the same, the smaller the true number, the greater the function value. （0

The logarithmic function properties are as follows:1. Definition domain solving: The definition domain of the logarithmic function y=logax is, but if you encounter the solution of the definition domain of the logarithmic composite function, in addition to paying attention to greater than 0, you should also pay attention to the fact that the base number of the spine Xunsun is greater than 0 and is not equal to 1, such as finding the definition domain of the function y=logx(2x-1), you need to meet the requirements of x>0 and x≠1 and 2x-1>0 at the same time, and get x>1 2 and x≠1, that is, the definition domain of the cherry chain is.

2. Range: The set of real numbers r, obviously the logarithmic function is unbounded.

3. Fixed point: The function image of the logarithmic function is constant above the fixed point (1,0).

4. Monotonicity: When a>1 is used, it is a monotonic increment function in the defined domain.

6. Parity: non-odd and non-even functions.

7. Periodicity: It is not a periodic function.

8. Symmetry: none.

9. The most valuable Changhan: none.

10. Zero point: x=1.