Question

find the value of x

$\bigstar\boxed{ \sf \bigg( \dfrac{4}{9} \bigg) ^{ \sqrt{x} } = (2.25)^{\sqrt{x} - 4 } }$
Topic - Exponential Functions

Class - 11th ​

Answer 1

$find \: the \: value \: of \: x$

Question:-

$( \frac{4}{9} ) = 2.25 ^{ \sqrt{x - 4} }$

Solution:-

$( \frac{4}{9}) ^{x} = \frac{9}{4} \ ^{ \sqrt{} x - 4} \\ = \ - \sqrt{} {x} = \sqrt{x} - 4 = \sqrt{ x} \\ = 2 = x = 4$

Hope it will help you

Answer 2

$\large{ \pmb{ \sf solution : - }}$

$: \implies \sf \bigg( \dfrac{4}{9} \bigg) ^{ \sqrt{x} } = (2.25)^{\sqrt{x} - 4 }$

$: \implies \sf \bigg( \dfrac{4}{9} \bigg) ^{ \sqrt{x} } = \bigg( \dfrac{225}{100} \bigg) ^{ \sqrt{x} - 4 }$

$: \implies \sf \bigg( \dfrac{4}{9} \bigg) ^{ \sqrt{x} } = \bigg( \dfrac{9}{4} \bigg) ^{ \sqrt{x} - 4 }$

$: \implies \sf \bigg [\bigg( \dfrac{9}{4} \bigg) ^{( - 1)} \bigg] ^{ (\sqrt{x}) } = \bigg( \dfrac{9}{4} \bigg) ^{( \sqrt{x} - 4 )}$

$: \implies \sf \bigg( \dfrac{9}{4} \bigg) ^{ - \sqrt{x} } = \bigg( \dfrac{9}{4} \bigg) ^{ \sqrt{x} - 4}$

Since the base are equal in both the sides we can equate their exponents !

$: \rightarrow \sf - \sqrt{x} = \sqrt{x} - 4$

$: \rightarrow \sf \sqrt{x} = 2$

$: \rightarrow \sf x = 4$