QUIZ #322a log x = x log alog x = 2Nilai x + a = ...?.... Question from @Elderberry

Rixxel17

log x = 2

x = 10²

x = 100

$\: \:$

^a log x = ^x log a

^a log 100 = ^100 log a

a = 100

$\: \:$

x + a

= 100 + 100

= 200

0 votes Thanks 1

henriyulianto

Ada 2 kemungkinan nilai $x+a$ , yaitu:

$\large\text{$\begin{aligned}\boxed{\ {\bf100}{\ \ \sf atau\ \ }{\bf100\,\tfrac{1}{100}}\ }\end{aligned}$}$

Pembahasan

Logaritma

Diketahui

$\begin{aligned}\bullet\ \ &{}^a\log x={}^x\log a\\\bullet\ \ &\log x=2\end{aligned}$

Ditanyakan

$x+a={\dots}?$

PENYELESAIAN

$\large\text{$\begin{aligned}&{}^a\log x={}^x\log a\\&{\Rightarrow\ }{}^a\log x=\frac{1}{{}^a\log x}\\&\textsf{Ambil }u={}^a\log x:\\&{\Rightarrow\ }u=\frac{1}{u}\\&{\Rightarrow\ }u^2=1\\&{\Rightarrow\ }u={}\pm1\end{aligned}$}$

Jika $u = \bf1$ , maka:

$\large\text{$\begin{aligned}&{}^a\log x=1\\&{\Rightarrow\ }\frac{\log x}{\log a}=1\\&{\Rightarrow\ }\log a=\log x\\&{\Rightarrow\ }a=x=10^{\log x}=10^2=100\\&{\Rightarrow\ }a+x=\bf200\end{aligned}$}$

Jika $u=\bf{-}1$ , maka:

$\large\text{$\begin{aligned}&{}^a\log x=-1\\&{\Rightarrow\ }\frac{\log x}{\log a}=-1\\&{\Rightarrow\ }\log a=-\log x\\&{\Rightarrow\ }\log a=\log x^{-1}\\&{\Rightarrow\ }\begin{cases}x=10^{\log x}=10^2=100\\a=x^{-1}=\frac{1}{100}\end{cases}\\&{\Rightarrow\ }a+x=100+\frac{1}{100}\\&{\Rightarrow\ }a+x=\bf100\,\frac{1}{100}=\frac{10001}{100}\end{aligned}$}$

Pemeriksaan

Untuk kedua kasus di atas, periksa apakah benar ${}^a\log x={}^x\log a$ .

Untuk a = x = 100, tentu benar karena basis dan numerus logaritma bernilai sama.

Untuk x = 100 dan a = 1/100:
$\large\text{$\begin{aligned}{}^a\log x&={}^{\frac{1}{100}}\log100\\&={}^{100^{-1}}\log100\\&=\frac{1}{-1}\cdot{}^{100}\log 100\\&=(-1)\cdot{}^{100}\log 100\\&={}^{100}\log100^{-1}\\{}^a\log x&={}^x\log a\end{aligned}$}$

KESIMPULAN

∴ Dengan demikian, ada 2 kemungkinan nilai $x+a$ , yaitu:

$\large\text{$\begin{aligned}\boxed{\ {\bf100}{\ \ \sf atau\ \ }{\bf100\,\tfrac{1}{100}}\ }\end{aligned}$}$

1 votes Thanks 2

Pembahasan

Logaritma

KESIMPULAN

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