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Jika diketahui 2log 3 = a dan 3log 5 = b. Tentukan 15log 20 ?

Pertanyaan

Jika diketahui log presuperscript 2 space 3 space equals space a dan begin mathsize 14px style log presuperscript 3 space 5 space equals space straight b end style. Tentukan begin mathsize 14px style log presuperscript 15 space 20 end style ?

N. Puspita

Master Teacher

Jawaban terverifikasi

Pembahasan

Ingat sifat logaritma begin mathsize 14px style log presuperscript a space b space equals space fraction numerator 1 over denominator log presuperscript b space a end fraction end style berarti

begin mathsize 14px style log presubscript blank presuperscript 3 space 2 space equals space fraction numerator 1 over denominator log presuperscript 2 space 3 end fraction equals 1 over a end style  

begin mathsize 14px style log presuperscript 15 space 20 end style = begin mathsize 14px style fraction numerator log presuperscript 3 space 20 over denominator log presuperscript 3 space 15 end fraction end style   

                 = begin mathsize 14px style fraction numerator log presuperscript 3 space open parentheses 5 cross times 2 cross times 2 close parentheses over denominator log presuperscript 3 space open parentheses 5 cross times 3 close parentheses end fraction end style  

                 = begin mathsize 14px style fraction numerator log presuperscript 3 space 5 space plus space 2 cross times log presuperscript 3 space 2 over denominator log presuperscript 3 space 5 space plus space log presuperscript 3 space 3 end fraction end style   

                 begin mathsize 14px style equals fraction numerator b plus begin display style 2 over a end style over denominator b plus 1 end fraction end style     

                 = begin mathsize 14px style fraction numerator a b plus 2 over denominator a b plus a end fraction end style  

                  = begin mathsize 14px style fraction numerator a b plus 2 over denominator a open parentheses b plus 1 close parentheses end fraction end style  

Jadi begin mathsize 14px style log presuperscript 15 space 10 end style = begin mathsize 14px style fraction numerator a b plus 2 over denominator a open parentheses b plus 1 close parentheses end fraction end style .

 

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