Dosen Jurusan EP_FE_UP

Senin, 02 September 2013

PANGKAT, AKAR DAN LOGARITMA


Rumus Bentuk Pangkat, Akar, dan Logaritma

a. Bentuk Pangkat
1). Jika n bilangan bulat positif dan a bilangan real maka a pangkat n didefinisikan sebagai berikut:
 
http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20a%5E%7Bn%7D=%5Cunderbrace%7Ba%20%5C:%20%5C:%20x%20%5C:%20%5C:%20a%5C:%5C:%20x%5C:%20....x%5C:%20a%7D

Pengertian pangkat tersebut diperluas, yaitu untuk   http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20a%5Cneq%200  berlaku:

http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20a%29%5C:%20%5C:%20a%5E%7B0%7D=1


http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20b%29%5C:%20a%5E%7B-n%7D=%5Cfrac%7B1%7D%7Ba%5E%7Bn%7D%7D

2). Sifat-sifat pengerjaan hitung bilangan berpangkat

http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20a%29.%5C:%20%5C:%20a%5E%7Bm%7D%5C:%20x%5C:%20a%5E%7Bn%7D=a%5E%7Bm+n%7D

 http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20b%29.%5C:%20%5C:%20%5Cfrac%7Ba%5E%7Bm%7D%7D%7Ba%5E%7Bn%7D%7D=a%5E%7Bm-n%7D

 http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20c%29.%5C:%20%5C:%20%28a%5E%7Bm%7D%29%5E%7Bn%7D=a%5E%7Bm.n%7D

 http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20d%29.%5C:%20%5C:%20%28a%5C,%20x%5C,%20b%29%5E%7Bn%7D=a%5E%7Bn%7D%5C,%20x%5C,%20b%5E%7Bn%7D

 http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20e%29.%5C:%20%5C:%20%5Cleft%20%28%20%5Cfrac%7Ba%7D%7Bb%7D%20%5Cright%20%29%5E%7Bn%7D=%5Cfrac%7Ba%5E%7Bn%7D%7D%7Bb%5E%7Bn%7D%7D%5C,%20%5C,%20%5C:%20b%5Cneq%200

b. Bentuk Akar
1). Jika a dan b bilangan real serta n bilangan bulat positif, maka:

      http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20a%29.%5C:%20%5C:%20b%5E%7Bn%7D=a%5C,%20%5CLeftrightarrow%20%5C,%20%5Csqrt%5bn%5d%7Ba%7D=b

       http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20b%29.%5C:%20%5C:%20%5Csqrt%5bn%5d%7Ba%7D=a%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D

 2). Sifat-sifat bentuk akar.
       http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20a%29.%5C:%20%5C:%20%5Csqrt%5bn%5d%7Ba%5E%7Bm%7D%7D=a%5E%7B%5Cfrac%7Bm%7D%7Bn%7D%7D
     
       http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20b%29.%5C:%20%5C:%20p%5Csqrt%5bn%5d%7Ba%7D+q%5Csqrt%5bn%5d%7Ba%7D=%28p+q%29%5Csqrt%5bn%5d%7Ba%7D

       http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20c%29.%5C:%20%5C:%20p%5Csqrt%5bn%5d%7Ba%7D-q%5Csqrt%5bn%5d%7Ba%7D=%28p-q%29%5Csqrt%5bn%5d%7Ba%7D
       http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20d%29.%5C:%20%5C:%20%5Csqrt%5bn%5d%7Ba.b%7D=%5Csqrt%5bn%5d%7Ba%7D%5C,%5C,%20x%5C,%20%5Csqrt%5bn%5d%7Bb%7D
     
       http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20e%29.%5C:%20%5C:%20%5Csqrt%5bn%5d%7B%5Cfrac%7Ba%7D%7Bb%7D%7D=%5Cfrac%7B%5Csqrt%5bn%5d%7Ba%7D%7D%7B%5Csqrt%5bn%5d%7Bb%7D%7D%5C,%20,b%5Cneq%20%5Cneq%200
     
      http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20f%29.%5C:%20%5C:%20%5Csqrt%5bm%5d%7B%5Csqrt%5bn%5d%7Ba%7D%7D=%5Csqrt%5bm.n%5d%7Ba%7D
     
     
     
3). Merasionalkan penyebut pecahan bentuk akar

http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20a%29.%5C:%20%5C:%20%5Cfrac%7Ba%7D%7B%5Csqrt%7Bb%7D%7D=%5Cfrac%7Ba%7D%7B%5Csqrt%7Bb%7D%7D%5C,%20x%5C,%20%5Cfrac%7B%5Csqrt%7Bb%7D%7D%7B%5Csqrt%7Bb%7D%7D=%5Cfrac%7Ba%7D%7Bb%7D%5Csqrt%7Bb%7D

             http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20=%5Cfrac%7Ba%7D%7Bb-c%7D%28%5C,%20%5Csqrt%7Bb%7D-%5Csqrt%7Bc%7D%5C,%20%5C,%20%29


c. Logaritma
1). Jika a dan b bilangan positif dengan  http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20p%5Cneq%201  maka berlakut:
http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20%7B%7D%5E%7Bp%7D%5Clog%7Ba%7D=n%5C,%20%5CLeftrightarrow%20%5C,%20p%5E%7Bn%7D=a

Dari hubungan tersebut, diperoleh:
http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20a%29.%5C,%20%5C,%20p%5E%7B0%7D=1%5C,%20%5CLeftrightarrow%20%5C,%20%7B%7D%5E%7Bp%7D%5Clog%7B1%7D=0
http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20b%29.%5C,%20%5C,%20p%5E%7B1%7D=p%5C,%20%5CLeftrightarrow%20%5C,%20%7B%7D%5E%7Bp%7D%5Clog%7Bp%7D=1
http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20c%29.%5C,%20%5C,%20p%5E%7Bn%7D=p%5E%7Bn%7D%5C,%20%5CLeftrightarrow%20%5C,%20%7B%7D%5E%7Bp%7D%5Clog%7Bp%5E%7Bn%7D%7D=n

2). Sifat-sifat logaritma
http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20a%29.%5C,%20%5C,%20%7B%7D%5E%7Bp%7D%5Clog%7Bab%7D=%5C,%20%7B%7D%5E%7Bp%7D%5Clog%7Ba%7D%5C,%20+%5C,%20%7B%7D%5E%7Bp%7D%5Clog%7Bb%7D
http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20b%29.%5C,%20%5C,%20%7B%7D%5E%7Bp%7D%5Clog%7B%5Cleft%20%28%20%5Cfrac%7Ba%7D%7Bb%7D%20%5Cright%20%29%7D=%5C,%20%7B%7D%5E%7Bp%7D%5Clog%7Ba%7D%5C,%20-%5C,%20%7B%7D%5E%7Bp%7D%5Clog%7Bb%7D
http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20c%29.%5C,%20%5C,%20%7B%7D%5E%7Bp%7D%5Clog%7Ba%5E%7Bn%7D%7D=%5C,%20n%5C,%20x%5C,%20%7B%7D%5E%7Bp%7D%5Clog%7Ba%7D
http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20d%29.%5C,%20%5C,%20%7B%7D%5E%7Bp%7D%5Clog%7Ba%7D=%5C,%20%5Cfrac%7B%7B%7D%5E%7Bn%7D%5Clog%7Ba%7D%7D%7B%7B%7D%5E%7Bn%7D%5Clog%7Bp%7D%7D
http://latex.codecogs.com/png.latex?%5Cinline%20%5Cfn_cm%20e%29.%5C,%20%5C,%20p%5E%7B%7B%7D%5E%7Bp%7D%5Clog%7Ba%7D%7D=a

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