log-5-4x-log-10-x-find-x-

Question Number 97725 by john santu last updated on 09/Jun/20

\log_{5} (4 x) = \log_{10} (x)

find x ?

Commented by prakash jain last updated on 09/Jun/20

4 x = 5^{y}

x = 10^{y}

4 \times 10^{y} = 5^{y}

2^{y} = \frac{1}{4} \Rightarrow y = - 2

x = \frac{1}{100}

Commented by bobhans last updated on 09/Jun/20

\frac{\ln (4 x)}{\ln (5)} = \frac{\ln (x)}{\ln (10)}

\frac{2 \ln (2) + \ln (x)}{\ln (5)} = \frac{\ln (x)}{\ln (5) + \ln (2)}

\ln (5) \ln (x) = (2 \ln (2) + \ln (x)) (\ln (5) + \ln (2))

\ln (5) \ln (x) = 2 \ln (5) \ln (2) + {2 \ln}^{2} (2) + \ln (5) \ln (x) + \ln (x) \ln (2)

0 = 2 \ln (5) \ln (2) + {2 \ln}^{2} (2) + \ln (x) \ln (2)

0 = 2 \ln (5) + 2 \ln (2) + \ln (x)

\ln (x) = - 2 \ln (10) = \ln (10^{- 2})

\Leftrightarrow x = \frac{1}{100}

Commented by john santu last updated on 09/Jun/20

thank both