How to find all the formulas about logarithms, how to find logarithmic formulas?

Denote the power with , and log(a)(b) denote the logarithm of b with a base.

denotes the multiplication sign, denotes the division sign.

Definition: If a n=b(a>0 and a≠1).

then n=log(a)(b).

Basic Properties:; Derivation 1This doesn't need to be pushed, it can be obtained directly from the definition (bring [n=log(a)(b)] in the definition to a n=b).

mn=m*n

by basic properties 1 (replace m and n).

a^[log(a)(mn)] = a^[log(a)(m)] a^[log(a)(n)]

by the nature of the exponent.

a^[log(a)(mn)] = a^

And because the exponential function is a monotonic function, so.

log(a)(mn) = log(a)(m) +log(a)(n)

3.Similar treatment to 2.

mn=m/n

by basic properties 1 (replace m and n).

a^[log(a)(m/n)] = a^[log(a)(m)] / a^[log(a)(n)]

by the nature of the exponent.

a^[log(a)(m/n)] = a^

And because the exponential function is a monotonic function, so.

log(a)(m/n) = log(a)(m) -log(a)(n)

4.Similar treatment to 2.

m^n=m^n

by the basic property 1 (replace m).

a^[log(a)(m^n)] = ^n

by the nature of the exponent.

a^[log(a)(m^n)] = a^

And because the exponential function is a monotonic function, so.

log(a)(m^n)=nlog(a)(m)

Other properties: Nature 1: Bottom change formula.

log(a)(n)=log(b)(n) / log(b)(a)

The derivation is as follows: n = a [log(a)(n)].

a = b^[log(b)(a)]

A combination of two types is available.

n = ^[log(a)(n)] = b^

And because n=b [log(b)(n)].

So b [log(b)(n)] = b

So log(b)(n) = [log(a)(n)]*log(b)(a)]].

So log(a)(n) = log(b)(n) log(b)(a).

Nature 2: (I don't know what the name is).

log(a^n)(b^m)=m/n*[log(a)(b)]

It is derived from the formula [lnx is log(e)(x) and e is called the base of the natural logarithm].

log(a^n)(b^m)=ln(a^n) / ln(b^n)

It can be obtained from basic properties 4.

log(a^n)(b^m) = [n*ln(a)] / [m*ln(b)] = (m/n)*

Then by the bottom change formula.

log(a^n)(b^m)=m/n*[log(a)(b)]

If a x=n(a>0, and a≠1), then x is called the logarithm of n with the loose wheel a as the base, denoted as x=log(a)(n), where a should be written at the bottom right of log. where a is called the base of the logarithm and n is called the true number.

Usually we refer to the logarithm with a base of 10 as the common logarithm, and the logarithm with the base of e as the natural logarithm.

In mathematics, logarithm is the inverse of exponentiation, just as division is the reciprocal of multiplication, and vice versa.

This means that the logarithm of one number is the exponent that must produce an imprint to another fixed number (cardinality). In the simple case, the logarithmic count factor in the multiplier.

The basic logarithmic formula is: x=log(a)(n).

The logarithmic formula is a common formula in mathematics, if a x = n (a>0, and a ≠ 1), then x is called the logarithm with a base n, usually we call the logarithm with a base of 10 as the common logarithm, and the logarithm with e as the base is called the natural logarithm.

If a x = n (a>0 and a is not equal to 1), then the number x is called the logarithm of n with a as the base, denoted as x=log(a)(n), where a should be written at the bottom right of log. The logarithmic properties and algorithms are as follows. loga（1）=0；loga（a）=1；Negative Zen calendar with zero has no logarithm, and a logan=n(a>0,a≠1).

Ask for the number of servants (xlogax).'=logax+1 lna, where a in logax is the base number and x is the true number; （logax）'=1 xlna, which is special i.e. a=e, is present (logex).'=lnx）'=1/x。

Derivation of the formula for changing the bottom: Let e x=b m, e y=a n then log(e y)(b m)=log(e y)(e x)=x y x=ln(b m),y=ln(a n) obtain: log(a n)(b m)=ln(b m) ln(a n).

The algorithm of the logarithmic formula, as shown in the following figure:

The derivation process is as follows:

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) (n r) (4)log(a n)(m)=1 nlog(a)(m)(n r) (5) Bottom change formula: log(a)m=log(b)m log(b)a (b>0 and b≠1) (6)a (log(b)n)=n (log(b)a) proven:

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) (7) logarithmic identity: a log(a)n=n; log(a)a b=b (8) is derived from the logarithmic properties of the power (derive the formula) , log(a)m (-1 n)=(-1 n)log(a)m , log(a)m (-m n)=(-m n)log(a)m , log(a n)m m = (m n)log(a)m base a at n root (m at n root is the true number) = log(a)m , log (base a at n root) (m at m root number is true) = (n m) log(a) m

The relationship between logarithms and exponents.

When a>0 and a≠1, a x=n x= (a)n take your time.

The arithmic nature of logarithms.

When a>0 and a≠1, m>0, n>0, then envy:

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)log(a^n)(m)=（1/n）log(a)(m)(n∈r)

5) Bottom change formula: log(a)m=log(b)m log(b)a (b>0 and b≠1).

6)a^(log(b)n)=n^(log(b)a)

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).

7) logarithmic identity: a log(a)n=n;

log(a)a b=b proof: let a log(a)n=x, log(a)n=log(a)x, n=x

8) The equation can be derived from the properties of the logarithm of the power

log(a)m^(-1/n)=(1/n)log(a)m

log(a)m^(-m/n)=(m/n)log(a)m

log(a^n)m^m=(m/n)log(a)m

Take a under the nth root as the base) (m under the n root as the true number) = log(a)m, log (base a under the n root number) (m under the m root as the true number) = (n m)log(a)m