Pembahasan Soal UN Materi Logaritma

Konsep dasar yang digunakan dalam perhitungan:

• $^{a}log\ b+^{a}log\ c=^{a}log\ b.c$
• $^{a}log\ b-^{a}log\ c=^{a}log\ \left ( \frac{b}{c} \right )$
• $^{a}log\ b\ ^{b}log\ c=^{a}log\ c$
• $^{a^{n}}log\ b^{m}=\frac{m}{n}\ ^{a}log\ b$

Soal 1. UN SMA Tapel 2014/2015 Program Studi IPA
Hasil $\frac{^{7}log\ 16\sqrt{2}\ ^{2\sqrt{2}}log\ \frac{1}{49}+^{2}log\ \frac{1}{16}}{^{5}log\ 5\sqrt{5}+^{2}log\ 25\sqrt{5}}$ adalah ...
A. $10$
B. $\frac{5}{2}$
C. $\frac{-7}{2}$
D. $\frac{-5}{2}$
E. $-10$

Pembahasan
$\frac{^{7}log\ 16\sqrt{2}\ ^{2\sqrt{2}}log\ \frac{1}{49}+^{2}log\ \frac{1}{16}}{^{5}log\ 5\sqrt{5}+^{2}log\ 25\sqrt{5}}$

$=\frac{^{7}log\ 2^{4}.2^{\frac{1}{2}}\ ^{2.2^{\frac{1}{2}}}log\ 7^{-2}+^{2}log\ 2^{-4}}{^{5}log\ 125.5}$
$=\frac{^{7}log\ 2^{\frac{9}{2}}\ ^{2^{\frac{3}{2}}}log\ 7^{-2}+^{2}log\ 2^{-4}}{^{5}log\ 5^{4}}$
$=\frac{\left ( \frac{9}{2} \right )\left ( \frac{1}{\frac{3}{2}} \right )(-2)\ ^{7}log\ 2\ ^{2}log\ 7+(-4)\ ^{2}log\ 2}{4\ ^{5}log\ 5}$
$=\frac{(3)(-2)\ ^{7}log\ 7+(-4).1}{4.1}$
$=\frac{(-6)+(-4)}{4}$
$=\frac{(-10)}{4}$
$=\frac{-5}{2}$

Jawaban D

Soal 2. UN SMA Tapel 2016/2017 Program Studi IPS
Hasil dari $^{7} log 4.\ ^{2} log 5+ ^{7}log \frac{49}{25}=$ ...
A. $1$
B. $2$
C. $3$
D. $4$
E. $5$

Pembahasan
$^{7} log 4.\ ^{2} log 5+ ^{7}log \frac{49}{25}$

$= ^{7} log 2^{2}.\ ^{2} log 5+ ^{7}log \frac{7^{2}}{5^{2}}$
$= 2.\ ^{7} log 2.\ ^{2} log 5+ ^{7}log \left (\frac{7}{5} \right )^{2}$
$= 2.\ ^{7} log 5+ 2.\ ^{7}log \left (\frac{7}{5} \right )$
$= 2. \left ( ^{7} log 5+\ ^{7}log \left (\frac{7}{5} \right ) \right )$
$= 2.\ ^{7} log\ 5.\frac{7}{5}$
$= 2.\ ^{7} log\ 7=2$

Jawaban B

Soal 3. UN SMA Tapel 2016/2017 Program Studi IPA
Hasil $\frac{^{\sqrt{3}} log\ 5\ ^{25} log\ 3\sqrt{3}-^{4} log\ 16}{^{3} log\ 54-^{3} log\ 2}$ adalah ...
A. $-\frac{9}{2}$
B. $-\frac{1}{6}$
C. $-\frac{1}{3}$
D. $3$
E. $\frac{9}{2}$

Pembahasan
$\frac{^{\sqrt{3}} log\ 5\ ^{25} log\ 3\sqrt{3}-^{4} log\ 16}{^{3} log\ 54-^{3} log\ 2}$

$=\frac{^{3^{\left ( \frac{1}{2} \right )}} log\ 5\ ^{5^{5}} log\ 3^{\left ( \frac{3}{2} \right )}-^{4} log\ 4^{2}}{^{3} log\ \frac{54}{2}}$
$=\frac{\frac{1}{\frac{1}{2}}.\frac{1}{2}.\frac{3}{2}\ ^{3} log\ 5\ ^{5} log\ 3-2.\ ^{4} log\ 4}{^{3} log\ 27}$
$=\frac{\frac{3}{2}\ ^{3} log\ 3-2.1}{^{3} log\ 3^{3}}$
$=\frac{\frac{3}{2}\ ^{3} log\ 3-2.1}{3.\ ^{3} log\ 3}$
$=\frac{\frac{3}{2}.1-2.1}{3.1}$
$=\frac{\frac{3}{2}-2}{3}$
$=-\frac{1}{6}$

Jawaban B