∫(c z)x = b, which leads to c (zx) = b, that is, y = z x.
Accord to that definitions of exponent and logarithm,
The formula for changing the bottom is x=y/z, which has been proved. 2, the form of the formula for changing the base: the formula for changing the base is an important formula, which is used in many logarithmic calculations and is also the focus of high school mathematics. Logs (a) and (b) represent logarithms based on b. The so-called bottoming formula is the derivation process of log(a)(b)=log(n)(b)/log(n)(a): If there is a logarithmic log(a)(b), let a = n x, b = n y (n > 0, n is not 1), Then log (a) (b) = log (n x) (n y) According to the basic formula log (a) (m n) = nloga (m) and the basic formula log (a n) m = 1 Then there is: log (a) (b) = log (n x) (n y). The application of base exchange formula: 1. Usually, when dealing with mathematical operations, the common base is converted into the common logarithm (in) with the base of E or the common logarithm (lg) with the base of 10, which is convenient for our operations. Sometimes use the formula to prove or solve related problems; 2. In engineering technology, the formula of changing bases is also commonly used. For example, in programming languages, some programming languages (such as C language) do not have logarithmic functions with A as the base and B as the real number; There are only commonly used logarithms e or 10 (i.e. In, Ig). At this time, it is necessary to replace logarithms with e or 10 to express the logarithmic expression of real numbers with b as the base, so as to deal with some practical problems. 4. The so-called bottoming formula is log(a)(b)=log(n)(b)/log(n)(a).
The derivation process of the bottom-changing formula;
If there is logarithm log(a)(b), let a = n x and b = n y.
Then log (a) (b) = log (n x) (n y)
According to the basic logarithmic formula log (a) (m n) = nlog (a) (m)
And the basic formula log (a n) (m) = 1/n× log (a) (m).
Simple logarithm (n x) (n y) = y/x
X = log (n) (a) and y = log (n) (b) can be obtained from a = n x and b = n y.
Then there is: log (a) (b) = log (n x) (n y) = log (n) (b)/log (n) (a).
Prove: log (a) (b) = log (n) (b)/log (n) (a) .5, let logab = K.
So a k = b
Because logc b=logc a^k=klogc a k = klogca k = klogca.
So (logc b)/(logc a)=k=loga b 6, let a = m square of x and b = n square of x, then log (a) b = log (m square of x)) =M/N)*log(a)b,
Then bring m = log (x) a and n = log (x) b back to m/n.
M = log (x) a, n = log (x) b because a = m of x and b = n of x.
7, the underlying formula
log(a)(N)=log(b)(N) / log(b)(a)
Derived as follows
N = a^[log(a)(N)]
a = b^[log(b)(a)]
By combining the two formulas, it can be concluded that.
n = {b^[log(b)(a)]}^[log(a)(n)]= b^{[log(a)(n)]*[log(b)(a)]}
And because n = b [log (b) (n)]
therefore
b^[log(b)(n)]= b^{[log(a)(n)]*[log(b)(a)]}
therefore
Log (b) (n) = [log (a) (n)] * [log (b) (a)] So log (a) (n) = log (b) (n)/log (b) (a) 8, n.
Let y = loga
y
Then a = n.
Take the logarithm with a base on both sides.
An n
ylogm =logm
ordinary
logm
y= -
a
logm
ordinary
Logarithmic meter
Which means loga =-
Answer.
logm
Let a b = n.............( 1)
Then b = Logan ............. (2)
Substitute ② into ① to obtain logarithmic identity:
a^(logaN)=N…………③
Take the logarithm of both sides with m as the base.
logaN logma=logmN
therefore
LogaN=(logmN)/(logma) 9, where N=alogaN and both sides take the logarithm based on b, logbn = logbalogan. logbalogan = Logan？ 6? 1logba，