Quiz(63/100)[tex] \: \: \: \: \: \: \: \: \: [/tex]Terlampir.... Question from @Aikey

Nilai dari $\sf{ \frac{ab}{a + b}}$ adalah 2.

=======================

PEMBAHASAN

Logaritma adalah salah satu konsep untuk menentukan nilai eksponen sehingga logaritma sering disebut sebagai kebalikan perpangkatan.

Ada berbagai macam sifat-sifat logaritma, yakni :

^a log b = c → a^c = b
^a log b = c → c = log b/log a
log a + log b = log (a × b)
log a - log b = log (a/b)
^a log b × ^b log c = ^a log c

→ DIKETAHUI :

3^a = 225
3^b = 225

→ DITANYA :

$\sf{ \frac{ab}{a + b}} =$

→ PENYELESAIAN :

Cara 1

$\sf{ {3}^{a} = {5}^{b} = 225}$

$\begin{gathered}\boxed{\begin{array}{lll}\bf {3}^{a} = 225 \\ \\ \bf {}^{3} \: log \: 225 = a \\ \\ \bf a = \frac{log \: 225}{log \: 3} \end{array}}\end{gathered}\begin{gathered}\boxed{\begin{array}{lll}\bf {5}^{b} = 225 \\ \\ \bf {}^{5} \: log \: 225 = b \\ \\ \bf b = \frac{log \: 225}{log \: 5} \end{array}}\end{gathered}$

$\sf{ \frac{ab}{a + b} = \frac{ \frac{log \: 225}{log \: 3}. \frac{log \: 225}{log \: 5}}{ \frac{log \: 225}{log \: 3} + \frac{log \: 225}{log \: 5}} }$

$\sf{ \frac{ab}{a + b} = \frac{ \frac{ \cancel{(log \: 225)}( \red{log \: 225})}{ \cancel{(log \: 3)(log \: 5)}}}{ \cancel{(log \: 225)} \: ( \frac{{ \red{log \: 3 + log \: 5}}}{ \cancel{(log \: 3)(log \: 5)}})}}$

$\sf{ \frac{ab}{a + b} = \frac{log \: 225}{ log \: 3 + log \: 5} }$

$\sf{ \frac{ab}{a + b} = \frac{log \: {15} {}^{2} }{ log \: 1 5} }$

$\sf{ \frac{ab}{a + b} = \boxed{ 2} }$

Cara 2

$\sf{3^{a} = 5^{b} = 225}$

$\sf{3^{a} = 5^{b} = 15^{2}}$

$\begin{gathered}\boxed{\begin{array}{lll}\bf {3}^{a} = 15^{2} \\ \\ \bf 3 = 15^{\frac{2}{a}}\end{array}}\end{gathered}$ $\begin{gathered}\boxed{\begin{array}{lll}\bf {5}^{b} = 15^{2} \\ \\ \bf 5 = 15^{\frac{2}{b}}\end{array}}\end{gathered}$