2 2 2 2 Logarithmic functions and their properties

If the nth power of a is equal to b (a is greater than 0 and a is not equal to 1), then the number n is called the logarithm of b with a as the base, denoted as n=loga to the power b, or log(a)b=n. where a is called the "base", b is called the "true number", and n is called the "logarithm of b with a as the base".

Correspondingly, the function y=logax is called a logarithmic function. The domain in which logarithmic functions are defined is (0,+ zero and negative numbers have no logarithms. The base number a is a constant, and its value range is (0,1) (1,+ Generally, when a=10, it is written: LGB=N.

Definition. If a n=b(a>0 and a≠1) then n=log(a)(b) is a fundamental property. If a>0, and a≠1, m>0, n>0, then

1、a^log(a)(b)=b

2、log(a)(a)=1

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)[m^（1/n）]=log(a)(m)/n7、logab*logba=1

Solution: Let's first look at the definition domain x+a 0, which makes sense when x 0, then if a 0 wants to be consistent with faction f(x) 0, that is, lg(x+a) 0 is constant.

The sufficient and necessary condition for the establishment of greater than 0 is that x+a 1 is constant, that is, when x 0, x 1-a is constant, and 1-a 0 is established

a 1 is a [1,+

f(x)+f(-x)

lg[√(x²+1)-x]+lg[√(x²+1)+x]=lg=lg(x²+1-x²)

lg10f(-x)=-f(x)

The domain of definition is r, symmetry with respect to the origin.

So it's an odd function.

Definition of logarithmic: In general, if a x = n (a>0 and a ≠ 1), then the number x is called the logarithm of a base n, denoted as x = logan, 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 as the independent variable, exponential as the dependent variable, and base number as a constant, which is called the logarithmic function.

where x is the independent variable and the domain of the function is (0,+ which is actually the inverse of the exponential function, which can be expressed as x=a y. 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].

Solving the domain of property definition: 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 number 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., its defined domain is .

Range: The set of real numbers r, which is obviously unbounded by logarithmic functions.

Fixed point: The function image is constant over the fixed point (1,0).

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

Logarithmic image.

00,a≠1，b>0）

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

When 01, y=logab<0;

When a>1, 00, a!=1---log a(x))'

lim(δx→0)((log a(x+δx)-log a(x))/δx)

lim(δx→0)(1/x*x/δx*log a((x+δx)/x))

lim(δx→0)(1/x*log a((1+δx/x)x/δx))

1/x*lim(δx→0)(log a((1+δx/x)x/δx))

1/x*log a(lim(δx→0)(1+δx/x)x/δx)

1/x*log a(e)

In particular, when a=e, (log a(x)).'=(ln x)'=1/x。

--Let y=ax take the logarithm ln y=xln a and find the x-derivative y on both sides'/y=ln ay'=yln a=a^xln a

Specifically, when a=e, y'=(ax)'=(ex)'=e^ln ex=ex。

lg[ [kx) 2+1]+kx], lg[ [kx) 2+1]-kx], ln[ [kx) 2+1]+kx], ln[ [kx) 2+1]-kx], are all odd functions. k=1 is lg[ [x 2+1]-x], and by definition, f(x)=-f(-x) is easy to prove.

When taking the college entrance examination, you use a lot, keep in mind.

If you have time, you can go to the math blog with Qingjie to watch "Odd and Even Functions in the College Entrance Examination".

Odd functions. If it's a multiple-choice question.