Question

find x if 9^x+2=26+3^2x+1​

Answer 1

Given :-

$\rm :\longmapsto\: {9}^{x + 2} = 26 + {3}^{2x + 1}$

To Find :-

• Value of x.

Solution :-

Given that,

$\rm :\longmapsto\: {9}^{x + 2} = 26 + {3}^{2x + 1}$

$\rm :\longmapsto\: {9}^{x} \times {9}^{2} = 26 + {3}^{2x} \times {3}^{1}$

$\rm :\longmapsto\: 81 \times {9}^{x} = 26 + 3 \times {9}^{x}$

$\rm :\longmapsto\: 81 \times {9}^{x} - 3 \times {9}^{x} = 26$

$\rm :\longmapsto\: {9}^{x}(81 - 3) = 26$

$\rm :\longmapsto\: {9}^{x} \times 78 = 26$

$\rm :\longmapsto\: {9}^{x} = \dfrac{26}{78}$

$\rm :\longmapsto\: {9}^{x} = \dfrac{1}{3}$

$\rm :\longmapsto\: {9}^{x} = {3}^{ - 1}$

$\rm :\longmapsto\: {3}^{2x} = {3}^{ - 1}$

$\rm :\longmapsto\:2x = - 1$

$\bf\implies \:x = - \: \dfrac{1}{2}$

Additional Information

Law of exponents :-

$\begin{gathered}(1)\:{\underline{\boxed{\bf{\blue{a^m\times{a^n}\:=\:a^{m\:+\:n}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(2)\:{\underline{\boxed{\bf{\purple{\dfrac{a^m}{a^n}\:=\:a^{m\:-\:n}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(3)\:{\underline{\boxed{\bf{\orange{\dfrac{1}{x^n}\:=\:x^{-n}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(4)\:{\underline{\boxed{\bf{\color{peru}{(a^m)^n\:=\:a^{m\times{n}}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(5)\:{\underline{\boxed{\bf{\red{ {x}^{0} = 1}}}}} \\ \end{gathered}$

$\begin{gathered}(6)\:{\underline{\boxed{\bf{\green{ {a}^{m} = {a}^{n} \implies \: m = n}}}}} \\ \end{gathered}$