Question

2^(x^2)=3^(x^2-4) let's solve​

Answer 1

Given:-

• ${2}^{ {x}^{2} } = {3}^{ ({x}^{2} - 4)}$

To find:-

• The value of x.

Answer:-

${2}^{ {x}^{2} } = {3}^{ ({x}^{2} - 4)}$

Taking log on both sides,

$\longrightarrow \log \big({2}^{ {x}^{2} } \big) = \log \big({3}^{ ({x}^{2} - 4)} \big)$

$\longrightarrow {x}^{2} \log \big({2} \big) = ( {x}^{2} - 4) \log \big({3}\big)$

$\longrightarrow \dfrac{ {x}^{2} - 4}{ {x}^{2} } = \dfrac{ \log(2)}{ \log(3)}$

$\longrightarrow \dfrac{ {x}^{2} }{ {x}^{2} } - \dfrac{4}{ {x}^{2} } = \dfrac{ \log(2)}{ \log(3)}$

$\longrightarrow 1 - \bigg({ \dfrac{2}{x} \bigg) }^{2} = \dfrac{ \log(2)}{ \log(3)}$

$\longrightarrow 1 - \dfrac{ \log(2)}{ \log(3)} = \bigg({ \dfrac{2}{x} \bigg) }^{2}$

$\longrightarrow \dfrac{ \log(3) - \log(2)}{ \log(3)} = \bigg({ \dfrac{2}{x} \bigg) }^{2}$

$\longrightarrow \dfrac{ \log(3/2) }{ \log(3)} = \bigg({ \dfrac{2}{x} \bigg) }^{2}$

Taking square root on both sides,

$\longrightarrow \sqrt{\dfrac{ \log(3/2) }{ \log(3)}} = \sqrt{\bigg({ \dfrac{2}{x} \bigg) }^{2}}$

$\longrightarrow \sqrt{\dfrac{ \log(3/2) }{ \log(3)}} = \pm \dfrac{2}{x}$

$\longrightarrow \pm\dfrac{1}{2} \sqrt{\dfrac{ \log(3/2) }{ \log(3)}} = \dfrac{1}{x}$

$\longrightarrow \pm 2 \sqrt{\dfrac{ \log(3) }{ \log(3/2)}} = x$

Hence,

${\longrightarrow \underline{ \underline{ x_{1} = 2 \sqrt{\dfrac{ \log(3) }{ \log(3/2)}} \approx 3.2921 }}}$

${\longrightarrow \underline{ \underline{ x_{2} = - 2 \sqrt{\dfrac{ \log(3) }{ \log(3/2)}} \approx - 3.2921 }}}$