Daftar isi
Pengertian Logaritma
Rumus-rumus dan Sifat-sifat Logaritma merupakan materi penting untuk memahami logaritma secara keseluruhan. Untuk itu, jangan pernah mengabaikan topik yang satu ini kalau ingin memahami logaritma. Kita langsung saja ke topik yang sesungguhnya. Logaritma adalah invers atau kebalikan dari eksponen atau perpangkatan. Di bab sebelumnya kita sudah membahas tentang eksponen atau perpangkatan.Sebagai contoh:
$2^5 = 32$, bentuk ini bisa kita nyatakan dalam bentuk logaritma yaitu $^2log\ 32 = 5$. Perhatikan: $3^{-2} = \dfrac{1}{9}$, bentuk ini dapat kita nyatakan dalam bentuk logaritma yaitu $^3log\ \dfrac{1}{9} = -2$. Jika $a^c = b$ maka $^alog\ b = c$ dimana $a > 0$ dan $a ≠ 1$ dan $b > 0$, $a$ disebut bilangan pokok logaritma. Jika $a = 10$, biasanya tidak ditulis. Jika bilangan pokoknya adalah $e$ dimana $(e = 2,718281828)$, maka logaritma ditulis dengan $ln$ (logaritma natural). $e$ disebut juga bilangan euler. $^elog\ b$ ditulis $ln\ b$. Adik-adik harus benar-benar memahami sifat-sifat logaritma, karena sifat-sifat logaritma inilah dasar kita untuk mengerjakan soal logaritma terutama soal-soal logaritma yang lebih rumit dan kompleks. Tanpa pemahaman yang baik tentang sifat-sifat logaritma, kita akan kesulitan untuk menghadapi soal-soal tentang logaritma.
Rumus-rumus dan Sifat-sifat Logaritma
$\boxed{I:\; Jika\; ^{a}log\ b = c → b = a^{c}}$Ilustrasi:
$a.\; Jika \;^2log\ 8 = 3$ maka $8 = 2^3$
$b.\; Jika \;^mlog\ n = p$ maka $n = m^p$
$\boxed{II:\; ^{a}log\ b = \dfrac{log\ b}{log\ a}}$
Ilustrasi:
$a.\; ^{2}log\ 5 = \dfrac{log\ 5}{log\ 2}$
$b.\; ^{m}log\ n = \dfrac{log\ n}{log\ m}$
$\boxed{III:\; ^{a}log\ bc =\ ^{a}log\ b + ^{a}log\ c}$
Ilustrasi:
$a.\; ^{3}log\ 2.7 =\ ^{3}log\ 2 +\ ^{3}log\ 7$
$b.\; ^{5}log\ 3.6 =\ ^{5}log\ 3 +\ ^{5}log\ 6$
$c.\; ^{2}log\ 14 =\ ^{2}log\ 2 +\ ^{2}log\ 7$
$\boxed{IV:\; ^{a}log\ \frac{b}{c} =\ ^{a}log\ b -\ ^{a}log\ c}$
Ilustrasi:
$a.\; ^{5}log\ \frac{3}{7} =\ ^{5}log\ 3 -\ ^{5}log\ 7$
$b.\; ^{2p}log\ \frac{3q}{4r} =\ ^{2p}log\ 3q -\ ^{2p}log\ 4r$
$\boxed{V:\; ^{a}log b^{m} = m.^{a}log b}$
Ilustrasi:
$a.\; ^{2}log\ 3^{5} =\ 5.^{2}log\ 3$
$b.\; ^{7}log 9^{4} =\ ^{7}log\ 3^8 =\ 8.^7log 3$
$\boxed{VI:\; ^{a}log\ a = 1}$
Ilustrasi:
$a.\; ^{5}log\ 5 = 1$
$b.\; ^{7}log\ 7 = 1$
$c.\; ^{pq}log\ pq = 1$
$\boxed{VII:\; ^{a}log\ 1 = 0}$
Ilustrasi:
$a.\; ^{6}log\ 1 = 0$
$b.\; ^{15}log\ 1 = 0$
$c.\; ^{5y}log\ 1 = 0$
$\boxed{VIII:\; ^{a}log\ b = \dfrac{1}{^{b}log\ a}}$
Ilustrasi:
$a.\; ^{3}log\ 7 = \dfrac{1}{^{7}log\ 3}$
$b.\; ^{5}log\ 11 = \dfrac{1}{^{11}log\ 5}$
$c.\; ^{x}log\ y = \dfrac{1}{^{y}log\ x}$
$\boxed{IX:\; ^{a}log\ b.^{b}log\ c =\ ^{a}log\ c}$
Ilustrasi:
$a.\; ^{2}log\ 3.^{3}log\ 7 = ^{2}log\ 7$
$b.\; ^{5}log\ 4.^{4}log\ 12 = ^{5}log\ 12$
$\boxed{X:\; ^{a^{m}}log\ b^{n} = \dfrac{n}{m}.^{a}log\ b}$
Ilustrasi:
$a.\; ^{3^{5}}log\ 2^{7} = \dfrac{7}{5}.^{3}log\ 2$
$b.\; ^{4^{9}}log\ 11^{15} = \dfrac{15}{9}.^{4}log\ 11$
$\boxed{XI:\; ^{a}log\ b = \dfrac{^{c}log b}{^{c}log\ a}}$
Ilustrasi:
$a.\; ^{3}log\ 5 = \dfrac{^{2}log\ 5}{^{2}log\ 3}$
$b.\; ^{15}log\ 18 = \dfrac{^{3}log\ 18}{^{3}log\ 15}$
Kita bisa mengganti $c$ dengan apa saja. Misalnya dengan angka $2,\ 3,\ 4,\ 5,\ p,\ q,\ r$ dan seterusnya.
$\boxed{XII:\; a^{^{a}log\ b} = b}$
Ilustrasi:
$a.\; 3^{^{3}log\ 5} = 5$
$b.\; 2^{^{2}log\ 7} = 7$
Contoh Soal Sifat-sifat Logaritma
$1.\ ^{3}log\ 81 = 4$ dapat ditulis sebagai . . . .
$A.\ 3^{2} = 9$
$B.\ 3^{3} = 27$
$C.\ 3^{4} = 81$
$D.\ 4^{2} = 16$
$E.\ 4^{3} = 64$
[Rumus dan sifat-sifat logaritma]
$A.\ 3^{2} = 9$
$B.\ 3^{3} = 27$
$C.\ 3^{4} = 81$
$D.\ 4^{2} = 16$
$E.\ 4^{3} = 64$
[Rumus dan sifat-sifat logaritma]
Ingat sifat logaritma:
Jika $^{a}log\ b = c → b = a^{c}$
$^{3}log\ 81 = 4$
$81 = 3^{4}$
$jawab:\ C.$
Jika $^{a}log\ b = c → b = a^{c}$
$^{3}log\ 81 = 4$
$81 = 3^{4}$
$jawab:\ C.$
$2.$ Jika $^{2}log\ 8 = x$, maka $^{x}log\ 81 =$ . . . .
$A.\ 2$
$B.\ 3$
$C.\ 4$
$D.\ 5$
$E.\ 6$
[Rumus dan sifat-sifat logaritma]
$A.\ 2$
$B.\ 3$
$C.\ 4$
$D.\ 5$
$E.\ 6$
[Rumus dan sifat-sifat logaritma]
Ingat sifat logaritma:
Jika $^{a}log\ b = c → b = a^{c}$.
$^{2}log\ 8 = x$
$8 = 2^{x}$
$2^{3} = 2^{x}$
$x = 3$
$^{x}log\ 81 =\ ^{3}log\ 81$
Ingat juga sifat logaritma:
$^{a}log\ b^{m} = m.^{a}log\ b$ dan sifat logaritma $^alog\ a = 1$.
Sehingga:
$^{x}log\ 81 =\ ^{3}log\ 81$
$=\ ^{3}log\ 3^{4}$
$= 4.^{3}log\ 3$
$= 4.1$
$= 4$
$jawab:\ C.$
Jika $^{a}log\ b = c → b = a^{c}$.
$^{2}log\ 8 = x$
$8 = 2^{x}$
$2^{3} = 2^{x}$
$x = 3$
$^{x}log\ 81 =\ ^{3}log\ 81$
Ingat juga sifat logaritma:
$^{a}log\ b^{m} = m.^{a}log\ b$ dan sifat logaritma $^alog\ a = 1$.
Sehingga:
$^{x}log\ 81 =\ ^{3}log\ 81$
$=\ ^{3}log\ 3^{4}$
$= 4.^{3}log\ 3$
$= 4.1$
$= 4$
$jawab:\ C.$
$3.$ Nilai dari $^{4}log\ 64 =$ . . . .
$A.\ 1$
$B.\ 2$
$C.\ 3$
$D.\ 4$
$E.\ 5$
[Rumus dan sifat-sifat logaritma]
$A.\ 1$
$B.\ 2$
$C.\ 3$
$D.\ 4$
$E.\ 5$
[Rumus dan sifat-sifat logaritma]
Ingat sifat logaritma:
$^{a}log\ b^{m} = m.^{a}log\ b$
$^{4}log\ 64 = ^{4}log\ 4^{3}$
$= 3. ^{4}log\ 4$
$= 3.1$
$= 3$
$jawab:\ C.$
$^{a}log\ b^{m} = m.^{a}log\ b$
$^{4}log\ 64 = ^{4}log\ 4^{3}$
$= 3. ^{4}log\ 4$
$= 3.1$
$= 3$
$jawab:\ C.$
$4.$ Nilai dari $^{2}log\ 4 -\ ^{2}log\ 6 +\ ^{2}log\ 12 =$ . . . .
$A.\ 0$
$B.\ 1$
$C.\ 2$
$D.\ 3$
$E.\ 4$
[Rumus dan sifat-sifat logaritma]
$A.\ 0$
$B.\ 1$
$C.\ 2$
$D.\ 3$
$E.\ 4$
[Rumus dan sifat-sifat logaritma]
Ingat sifat logaritma:
$^{a}log\ bc =\ ^{a}log\ b +\ ^{a}log\ c$ dan
$^{a}log\ \dfrac{b}{c} =\ ^{a}log\ b -\ ^{a}log\ c$
ingat juga sifat-sifat logaritma seperti pada soal nomor 1, 2 dan 3.
$^{2}log\ 4 -\ ^{2}log\ 6 +\ ^{2}log\ 12$
$=\ ^{2}log\ \dfrac{4.12}{6}$
$=\ ^{2}log\ 8$
$=\ ^{2}log\ 2^{3}$
$= 3.^{2}log\ 2$
$= 3.1$
$= 3$
$jawab:\ D.$
$^{a}log\ bc =\ ^{a}log\ b +\ ^{a}log\ c$ dan
$^{a}log\ \dfrac{b}{c} =\ ^{a}log\ b -\ ^{a}log\ c$
ingat juga sifat-sifat logaritma seperti pada soal nomor 1, 2 dan 3.
$^{2}log\ 4 -\ ^{2}log\ 6 +\ ^{2}log\ 12$
$=\ ^{2}log\ \dfrac{4.12}{6}$
$=\ ^{2}log\ 8$
$=\ ^{2}log\ 2^{3}$
$= 3.^{2}log\ 2$
$= 3.1$
$= 3$
$jawab:\ D.$
$5.\ ^{a}log\ bc -\ ^{a}log\ (\dfrac{b}{c})^{2} =$ . . . .
$A.\ ^{a}log\ (\dfrac{c^{3}}{b})$
$B.\ ^{a}log\ (\dfrac{b^{3}}{c})$
$C.\ ^{a}log\ (\dfrac{c^{2}}{b})$
$D.\ ^{a}log\ (\dfrac{c}{b^{3}})$
$E.\ ^{a}log\ (\dfrac{c}{b^{2}})$
[Rumus dan sifat-sifat logaritma]
$A.\ ^{a}log\ (\dfrac{c^{3}}{b})$
$B.\ ^{a}log\ (\dfrac{b^{3}}{c})$
$C.\ ^{a}log\ (\dfrac{c^{2}}{b})$
$D.\ ^{a}log\ (\dfrac{c}{b^{3}})$
$E.\ ^{a}log\ (\dfrac{c}{b^{2}})$
[Rumus dan sifat-sifat logaritma]
$^{a}log\ bc -\ ^{a}log\ (\dfrac{b}{c})^{2}$ $ =\ ^{a}log bc - (^{a}log\ b^{2} -\ ^{a}log\ c^{2})$
$=\ ^{a}log\ bc -\ ^{a}log\ b^{2} +\ ^{a}log\ c^{2}$
$=\ ^{a}log\ \dfrac{bc.c^{2}}{b^{2}}$
$=\ ^{a}log\ \dfrac{c^{3}}{b}$
$jawab:\ A.$
$=\ ^{a}log\ bc -\ ^{a}log\ b^{2} +\ ^{a}log\ c^{2}$
$=\ ^{a}log\ \dfrac{bc.c^{2}}{b^{2}}$
$=\ ^{a}log\ \dfrac{c^{3}}{b}$
$jawab:\ A.$
$6.$ Jika $log\ 5 = a$ dan $log\ 7 = b$, maka $log\ 175 =$ . . . .
$A.\ a + b$
$B.\ a - b$
$C.\ 2a + b$
$D.\ a + 2b$
$E.\ a - 2b$
[Rumus dan sifat-sifat logaritma]
$A.\ a + b$
$B.\ a - b$
$C.\ 2a + b$
$D.\ a + 2b$
$E.\ a - 2b$
[Rumus dan sifat-sifat logaritma]
$log\ 5 = a$ dan $log\ 7 = b$.
$log\ 175 = log\ 25.7$
$= log\ 25 + log\ 7$
$= log\ 5^{2} + log\ 7$
$= 2.log\ 5 + log\ 7$
$= 2a + b$
$jawab:\ C.$
$log\ 175 = log\ 25.7$
$= log\ 25 + log\ 7$
$= log\ 5^{2} + log\ 7$
$= 2.log\ 5 + log\ 7$
$= 2a + b$
$jawab:\ C.$
$7.\ ^{1/3}log\ 27 =$ . . . .
$A.\ 3$
$B.\ 2$
$C.\ 1$
$D.\ -2$
$E.\ -3$
[Rumus dan sifat-sifat logaritma]
$A.\ 3$
$B.\ 2$
$C.\ 1$
$D.\ -2$
$E.\ -3$
[Rumus dan sifat-sifat logaritma]
Ingat sifat logaritma:
$^{a^{m}}log\ b^{n} = \dfrac{n}{m}.^{a}log\ b$
$^{1/3}log\ 27 =\ ^{3^{-1}}log\ 3^{3}$
$=\ \dfrac{3}{-1}.^{3}log\ 3$
$= -3.^{3}log\ 3$
$= -3.1$
$= -3$
$jawab:\ E.$
$^{a^{m}}log\ b^{n} = \dfrac{n}{m}.^{a}log\ b$
$^{1/3}log\ 27 =\ ^{3^{-1}}log\ 3^{3}$
$=\ \dfrac{3}{-1}.^{3}log\ 3$
$= -3.^{3}log\ 3$
$= -3.1$
$= -3$
$jawab:\ E.$
$8.$ Jika $^{3}log^{2}log^{2}log\ x = 0$, maka $x =$ . . . .
$A.\ 0$
$B.\ 1$
$C.\ 2$
$D.\ 3$
$E.\ 4$
[Rumus dan sifat-sifat logaritma]
$A.\ 0$
$B.\ 1$
$C.\ 2$
$D.\ 3$
$E.\ 4$
[Rumus dan sifat-sifat logaritma]
$^{3}log^{2}log^{2}log\ x = 0$
$^{3}log^{2}log^{2}log\ x = ^{3}log\ 1$
$^{2}log^{2}log\ x = 1$
$^{2}log\ x = 2$
$x = 4$
$jawab:\ E.$
$^{3}log^{2}log^{2}log\ x = ^{3}log\ 1$
$^{2}log^{2}log\ x = 1$
$^{2}log\ x = 2$
$x = 4$
$jawab:\ E.$
$9.\ \dfrac{log x^{2}y + log\sqrt{xy} + logxy^{2}}{log\sqrt{xy}} =$ . . . .
$A.\ 1$
$B.\ 3$
$C.\ 5$
$D.\ 7$
$E.\ 9$
[Rumus dan sifat-sifat logaritma]
$A.\ 1$
$B.\ 3$
$C.\ 5$
$D.\ 7$
$E.\ 9$
[Rumus dan sifat-sifat logaritma]
$\dfrac{log x^{2}y + log\sqrt{xy} + logxy^{2}}{log\sqrt{xy}}$ $= \dfrac{log\ (x^2y.\sqrt{xy}.xy^2)}{log\ \sqrt{xy}}$
$= \dfrac{logx^{2}.x^{1/2}.x.y.y^{1/2}.y^{2}}{log (xy)^{1/2}}$
$= \dfrac{log(xy)^{7/2}}{log(xy)^{1/2}}$
$= \dfrac{7/2}{1/2}.\dfrac{log\ xy}{log\ xy}$
$= 7$
$jawab:\ D.$
$= \dfrac{logx^{2}.x^{1/2}.x.y.y^{1/2}.y^{2}}{log (xy)^{1/2}}$
$= \dfrac{log(xy)^{7/2}}{log(xy)^{1/2}}$
$= \dfrac{7/2}{1/2}.\dfrac{log\ xy}{log\ xy}$
$= 7$
$jawab:\ D.$
$10.$ Jika $log\ \left(\dfrac{a^{2}}{b^{4}}\right) = 6$, maka $log \sqrt[3]{\dfrac{b^{2}}{a}} =$ . . . .
A. -2
B. -1
C. 0
D. 1
E. 2
[Rumus dan sifat-sifat logaritma]
A. -2
B. -1
C. 0
D. 1
E. 2
[Rumus dan sifat-sifat logaritma]
$log\ \left(\dfrac{a^{2}}{b^{4}}\right) = 6$
$log\ \left(\dfrac{a}{b^{2}}\right)^{2} = 6$
$2.log\ \left(\dfrac{a}{b^{2}}\right) = 6$
$log\ \left(\dfrac{a}{b^{2}}\right) = 3$
$log\ a - log\ b^{2} = 3$
$log\ b^{2} - log\ a = -3$
$log\ \dfrac{b^{2}}{a} = -3$
$\dfrac{1}{3}.log\dfrac{b^{2}}{a} = -1$
$\log\sqrt[3]{\dfrac{b^{2}}{a}} = -1$
$jawab:\ B.$
$log\ \left(\dfrac{a}{b^{2}}\right)^{2} = 6$
$2.log\ \left(\dfrac{a}{b^{2}}\right) = 6$
$log\ \left(\dfrac{a}{b^{2}}\right) = 3$
$log\ a - log\ b^{2} = 3$
$log\ b^{2} - log\ a = -3$
$log\ \dfrac{b^{2}}{a} = -3$
$\dfrac{1}{3}.log\dfrac{b^{2}}{a} = -1$
$\log\sqrt[3]{\dfrac{b^{2}}{a}} = -1$
$jawab:\ B.$
$11.$ Jika $^{8}log\ 3 = a$ dan $^{3}log\ 5 = b$, maka $^{6}log\ 15 =$ . . . .
$A.\ \dfrac{3b(1 + a)}{3b + 1}$
$B.\ \dfrac{3a(1 + b)}{3a + 1}$
$C.\ \dfrac{3ab(1 + a)}{3b + 1}$
$D.\ \dfrac{3b(1 - a)}{3b + 1}$
$E.\ \dfrac{3a(1 - b)}{3a - 1}$
[Rumus dan sifat-sifat logaritma]
$A.\ \dfrac{3b(1 + a)}{3b + 1}$
$B.\ \dfrac{3a(1 + b)}{3a + 1}$
$C.\ \dfrac{3ab(1 + a)}{3b + 1}$
$D.\ \dfrac{3b(1 - a)}{3b + 1}$
$E.\ \dfrac{3a(1 - b)}{3a - 1}$
[Rumus dan sifat-sifat logaritma]
$^{8}log\ 3 = a$
$\displaystyle ^{2^{3}}log\ 3 = a$
$\dfrac{1}{3}.^{2}log\ 3 = a$
$^{2}log\ 3 = 3a$ . . . . *
$^{3}log\ 2 = \dfrac{1}{3a}$ . . . . **
$^{3}log\ 5 = b$ . . . . ***
$^{5}log\ 3 = \dfrac{1}{b}$ . . . . ****
$^{6}log\ 15 = \dfrac{log\ 15}{log\ 6}$
$= \dfrac {log\ 3.5}{log\ 3.2}$
$= \dfrac {log\ 3 + log\ 5}{log\ 3 + log\ 2}$
Tambahkan bilangan pokok $3$
$= \dfrac {^{3}log\ 3 + ^{3}log\ 5}{^{3}log\ 3 + ^{3}log\ 2}$
$= \dfrac {1 + b}{1 + \dfrac{1}{3a}}$
$= \dfrac {1 + b}{\dfrac{3a + 1}{3a}}$
$= \dfrac {3a(1 + b)}{3a + 1}$
$jawab:\ B.$
$\displaystyle ^{2^{3}}log\ 3 = a$
$\dfrac{1}{3}.^{2}log\ 3 = a$
$^{2}log\ 3 = 3a$ . . . . *
$^{3}log\ 2 = \dfrac{1}{3a}$ . . . . **
$^{3}log\ 5 = b$ . . . . ***
$^{5}log\ 3 = \dfrac{1}{b}$ . . . . ****
$^{6}log\ 15 = \dfrac{log\ 15}{log\ 6}$
$= \dfrac {log\ 3.5}{log\ 3.2}$
$= \dfrac {log\ 3 + log\ 5}{log\ 3 + log\ 2}$
Tambahkan bilangan pokok $3$
$= \dfrac {^{3}log\ 3 + ^{3}log\ 5}{^{3}log\ 3 + ^{3}log\ 2}$
$= \dfrac {1 + b}{1 + \dfrac{1}{3a}}$
$= \dfrac {1 + b}{\dfrac{3a + 1}{3a}}$
$= \dfrac {3a(1 + b)}{3a + 1}$
$jawab:\ B.$
$12.$ Nilai dari $\dfrac{1}{^{2}log 24} + \dfrac{1}{^{3}log 24} + \dfrac{1}{^{4}log 24} =$ . . . .
$A.\ 0$
$B.\ 1$
$C.\ 2$
$D.\ 3$
$E.\ 4$
[Rumus dan sifat-sifat logaritma]
$A.\ 0$
$B.\ 1$
$C.\ 2$
$D.\ 3$
$E.\ 4$
[Rumus dan sifat-sifat logaritma]
$\dfrac{1}{^{2}log\ 24} + \dfrac{1}{^{3}log\ 24} + \dfrac{1}{^{4}log\ 24}$
$=\ ^{24}log\ 2 +\ ^{24}log\ 3 +\ ^{24}log\ 4$
$=\ ^{24}log\ 2.3.4$
$=\ ^{24}log\ 24$
$= 1$
$jawab:\ B.$
$=\ ^{24}log\ 2 +\ ^{24}log\ 3 +\ ^{24}log\ 4$
$=\ ^{24}log\ 2.3.4$
$=\ ^{24}log\ 24$
$= 1$
$jawab:\ B.$
$13.$ Nilai dari $27^{\displaystyle ^{3}log\ 2} =$ . . . .
$A.\ 2$
$B.\ 4$
$C.\ 6$
$D.\ 8$
$E.\ 10$
[Rumus dan sifat-sifat logaritma]
$A.\ 2$
$B.\ 4$
$C.\ 6$
$D.\ 8$
$E.\ 10$
[Rumus dan sifat-sifat logaritma]
$27^{\displaystyle ^{3}log\ 2} = \left(3^{3}\right)^{\displaystyle {^{3}log\ 2}}$
$= 3^{\displaystyle {3.\displaystyle ^{3}log\ 2}}$
$= 3^{\displaystyle ^{3}log\ 8}$
$= 8$
$jawab:\ D.$
$= 3^{\displaystyle {3.\displaystyle ^{3}log\ 2}}$
$= 3^{\displaystyle ^{3}log\ 8}$
$= 8$
$jawab:\ D.$
$14.\ 4^{\displaystyle ^{2}log\ 3} + 2^{\displaystyle ^{4}log\ 9} =$ . . . .
$A.\ 9$
$B.\ 10$
$C.\ 11$
$D.\ 12$
$E.\ 13$
[Rumus dan sifat-sifat logaritma]
$A.\ 9$
$B.\ 10$
$C.\ 11$
$D.\ 12$
$E.\ 13$
[Rumus dan sifat-sifat logaritma]
$4^{\displaystyle ^{2}log\ 3} + 2^{\displaystyle ^{4}log\ 9}$
$= \left(2^2\right)^{\displaystyle ^{2}log\ 3} + \left(2\right)^{\frac12.\displaystyle^{2}log\ 9}$
$= \left(2\right)^{2.\displaystyle ^{2}log\ 3} + \left(2\right)^{\displaystyle^{2}log\ 9^{\frac12}}$
$= \left(2\right)^{\displaystyle ^{2}log\ 3^2} + \left(2\right)^{\displaystyle^{2}log\ 9^{\frac12}}$
$= 2^{\displaystyle ^{2}log\ 9} + 2^{\displaystyle ^{2}log\ \sqrt{9}}$
$= 9 + \sqrt{9}$
$= 9 + 3$
$= 12$
$jawab:\ D.$
$= \left(2^2\right)^{\displaystyle ^{2}log\ 3} + \left(2\right)^{\frac12.\displaystyle^{2}log\ 9}$
$= \left(2\right)^{2.\displaystyle ^{2}log\ 3} + \left(2\right)^{\displaystyle^{2}log\ 9^{\frac12}}$
$= \left(2\right)^{\displaystyle ^{2}log\ 3^2} + \left(2\right)^{\displaystyle^{2}log\ 9^{\frac12}}$
$= 2^{\displaystyle ^{2}log\ 9} + 2^{\displaystyle ^{2}log\ \sqrt{9}}$
$= 9 + \sqrt{9}$
$= 9 + 3$
$= 12$
$jawab:\ D.$
$15.\ \dfrac{log^{2}\ 7 - 2log\ 7.log\ 5 + log^{2}\ 5}{log\ \dfrac{7}{5}} =$ . . . .
$A.\ -1$
$B.\ 0$
$C.\ 1$
$D.\ log\ \dfrac{7}{5}$
$E.\ log\ \dfrac{5}{7}$
[Rumus dan sifat-sifat logaritma]
$A.\ -1$
$B.\ 0$
$C.\ 1$
$D.\ log\ \dfrac{7}{5}$
$E.\ log\ \dfrac{5}{7}$
[Rumus dan sifat-sifat logaritma]
$\dfrac{log^{2}\ 7 - 2log\ 7.log\ 5 + log^{2}\ 5}{log\ \dfrac{7}{5}}$
$= \dfrac{(log\ 7 - log\ 5)^{2}}{log\ 7 - log\ 5}$
$= log\ 7 - log\ 5$
$= log\ \dfrac{7}{5}$
$jawab:\ D.$
$= \dfrac{(log\ 7 - log\ 5)^{2}}{log\ 7 - log\ 5}$
$= log\ 7 - log\ 5$
$= log\ \dfrac{7}{5}$
$jawab:\ D.$
Di bawah ini ada soal tentang sifat-sifat logaritma untuk latihan. Silahkan adik-adik coba kerjakan. Tetap perhatikan sifat-sifat logaritma diatas supaya adik-adik bisa mengerjakannya.
Soal Latihan Sifat-sifat Logaritma
$1.$ Jika $a =\ ^{6}log\ 5$ dan $b =\ ^{5}log\ 4$, maka $^{4}log\ 0,24 =$ . . . .$A.\ \dfrac{a + 2}{ab}$
$B.\ \dfrac{2a + 1}{ab}$
$C.\ \dfrac{a - 2}{ab}$
$D.\ \dfrac{2a + 1}{2ab}$
$E.\ \dfrac{1 - 2a}{ab}$
$2.$ Jika $^{4}log\ 6 = m + 1$ , maka $^{9}log\ 8 =$ . . . .
$A.\ \dfrac{3}{4m - 2}$
$B.\ \dfrac{3}{4m + 2}$
$C.\ \dfrac{3}{2m + 4}$
$D.\ \dfrac{3}{2m - 4}$
$E.\ \dfrac{3}{2m + 2}$
$3.$ Jika $a > 1,\ b > 1,\ dan\ c > 1,$ maka
$^{b}log\ \sqrt{a}.^{c}log\ b^{2}.^{a}log\ \sqrt{c} =$ . . . .
$A.\ \dfrac{1}{4}$
$B.\ \dfrac{1}{2}$
$C.\ 1$
$D.\ 2$
$E.\ 3$
$4.$ Jika $^{4}log\ 5 = p$ dan $^{4}log\ 28 = q$, maka $^{4}log\ 70 =$ . . . .
$A.\ p + q - \dfrac{1}{2}$
$B.\ p + 2q + \dfrac{1}{2}$
$C.\ p - q + \dfrac{3}{2}$
$D.\ p - q + \dfrac{1}{2}$
$E.\ 2p - q + \dfrac{1}{2}$
$5.$ Nilai dari $^{3}log\ 4.^{2}log\ 125.^{5}log\ 81 =$ . . . .
$A.\ 24$
$B.\ 34$
$C.\ 44$
$D.\ 54$
$E.\ 64$
$6.$ Jika $^{2}log\ 3 = a$ dan $^{3}log\ 5 = b$, maka nilai dari
$^{15}log\ 32 =$ . . . .
$A.\ \dfrac{5}{a(1 + b)}$
$B.\ \dfrac{5}{b(1 + a)}$
$C.\ \dfrac{5}{ab(1 + b)}$
$D.\ \dfrac{5}{a(1 - ab)}$
$E.\ \dfrac{5}{a(1 - b)}$
$7.\ ^{a}log\ (\dfrac{1}{b}).^{b}log\ (\dfrac{1}{c}).^{c}log\ (\dfrac{1}{a}) =$ . . . .
$A.\ 1 - abc$
$B.\ 1 + abc$
$C.\ \dfrac{1}{abc}$
$D.\ -1$
$E.\ 1$
$8.$ Jika $b = a^{4}$, $a\ dan\ b$ positif, maka $^{a}log\ b -\ ^{b}log\ a =$ . . . .
$A.\ 0$
$B.\ 1$
$C.\ 2$
$D.\ 3\dfrac{3}{4}$
$E.\ 4\dfrac{1}{4}$
$9.$ Jika $^{2}log\ \dfrac{1}{a} = \dfrac{3}{2}$ dan $^{16}log\ b = 5$,
maka $^{a}log\ \dfrac{1}{b^{3}} =$ . . . .
$A.\ 40$
$B.\ -40$
$C.\ \dfrac{40}{3}$
$D.\ -\dfrac{40}{3}$
$E.\ 20$
$10.$ Jika $^{2}log\ x.^{5}log\ 4 = 6$, maka $x =$ . . . .
$A.\ 5^{12}$
$B.\ 5^{6}$
$C.\ 5^{3}$
$D.\ 2^{5}$
$E.\ 3^{5}$
Demikianlah dasar dan sifat-sifat logaritma, semoga bermanfaat.
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No. 9.... Ada yang keliru itu min
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