Misalkan diketahui 2log3=p dan 2log5=q, nyatakan tiap

a) $\frac{1+2q}{1+p}$, b) $\frac{2+q}{1+2p}$

Pembahasan

Untuk menyatakan ${^2}\log50$ dan ${^{18}}\log20$ dalam $p$ dan $q$, kita perlu menggunakan sifat-sifat logaritma.

Diketahui:
${^2}\log3 = p$
${^2}\log5 = q$

a) ${^6}\log50$
Kita ubah basis logaritma menjadi 2:
${^6}\log50 = \frac{{^2}\log50}{^2\log6}$
${^2}\log50 = {^2}\log(2 \times 5^2) = {^2}\log2 + {^2}\log5^2 = 1 + 2 \times {^2}\log5 = 1 + 2q$
${^2}\log6 = {^2}\log(2 \times 3) = {^2}\log2 + {^2}\log3 = 1 + p$
Jadi, ${^6}\log50 = \frac{1+2q}{1+p}$

b) ${^{18}}\log20$
Kita ubah basis logaritma menjadi 2:
${^{18}}\log20 = \frac{{^2}\log20}{^2\log18}$
${^2}\log20 = {^2}\log(2^2 \times 5) = {^2}\log2^2 + {^2}\log5 = 2 \times {^2}\log2 + {^2}\log5 = 2(1) + q = 2+q$
${^2}\log18 = {^2}\log(2 \times 3^2) = {^2}\log2 + {^2}\log3^2 = {^2}\log2 + 2 \times {^2}\log3 = 1 + 2p$
Jadi, ${^{18}}\log20 = \frac{2+q}{1+2p}$