If-log-x-y-6-amp-log-14x-8y-3-then-find-the-value-of-x-amp-y-

Question Number 70270 by Shamim last updated on 02/Oct/19

If \log_{x} y = 6 & \log_{14 x} 8 y = 3 then find the

value of x & y .

Answered by Rio Michael last updated on 02/Oct/19

{l o g}_{x} y = 6 \dots \dots \dots . . (1)

{l o g}_{14 x} 8 y = 3 \dots \dots \dots . (2)

f r o m e q n (1), y = x^{6} \dots \dots . (3)

f r o m e q n (2), 8 y = {(14 x)}^{3}

8 y = 2744 x^{3} \dots \dots . (4)

s u b s t i t u t e e q n (3) i n e q n (4)

\Rightarrow 8 x^{6} = 2744 x^{3}

8 x^{3} = 2744

\Rightarrow x = 7

x = 7 \Rightarrow y = 7^{6}

x = 7 a n d y = 7^{6}

Commented by Shamim last updated on 03/Oct/19

Here, x is base of logaritham & y is a

normal number . bt you mean that y is a

power of x .

Commented by Shamim last updated on 03/Oct/19

hmm . tnks

Commented by Rio Michael last updated on 03/Oct/19

y o u r w e l c o m s i r

Commented by MJS last updated on 03/Oct/19

\log_{x} y = \frac{\log y}{\log x} = 6

\log_{14 x} 8 y = \frac{\log y + 3 \log 2}{\log x + \log 7 + \log 2} = 3

\log y = 6 \log x

\log y = 3 (\log x + \log 7)

6 \log x = 3 (\log x + \log 7)

\log x = \log 7

x = 7

\log y = 6 \log 7

y = 7^{6}

Commented by MJS last updated on 03/Oct/19

(1) z = x^{y} \Leftrightarrow y = \log_{x} z

(2)

\log z = \log x^{y}

\log z = y \log x \Rightarrow y = \frac{\log z}{\log x}

\Rightarrow \log_{x} z = \frac{\log z}{\log x}

Commented by Shamim last updated on 03/Oct/19

tnks a lot . . bt one prblm

Commented by MJS last updated on 03/Oct/19

just change the names of the variables

z = \log_{x} y \Leftrightarrow x^{z} = y

\log x^{z} = \log y \Rightarrow z \log x = \log y \Rightarrow z = \frac{\log y}{\log x}

\Rightarrow \log_{x} y = \frac{\log y}{\log x}

Commented by Shamim last updated on 03/Oct/19

tnks a lot \dots

Commented by Shamim last updated on 03/Oct/19

plz give me the law - - -

\log_{x} y = \frac{\log y}{\log x} \dots . plz describe details .

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