Question

7*3^(x+1)-5^(x+2)=3^(x+4)-5^(x+3)

Answer 1

$\rm 7 \times 3^{x+1} - 5^{x+2} = 3^{x+4} - 5^{x+3}$

$\rm\implies 7 \times 3^{x+1} - 5^{x+1}\times 5 = 3^{x+1} \times 3^3- 5^{x+1}\times 5^2 \qquad ...[since , \ a^{m+n} = a^m \times a^n]$

$\implies\rm 7 \times 3^{x+1} - 5^{x+1}\times 5 = 3^{x+1} \times 27- 5^{x+1}\times 25$

Let $\rm 3^{x+1} = a \ and \ 5^{x+1} = b$

$\rm\implies 7 \times a - b \times 5 = a\times 27- b\times 25$

$\rm\implies 7a - 5b = 27a- 25b$

$\rm\implies 25b - 5b = 27a- 7a$

$\rm\implies 20b = 20a$

$\rm\implies b = a$

Substituting the values we get,

$\rm\implies 5^{x+1} = 3^{x+1}$

$\rm\implies 5^x \times 5= 3^{x} \times 3$

$\rm\implies 5^x = \dfrac{3^{x} \times 3}{5}$

$\rm\implies \dfrac{5^x}{3^x} = \dfrac{3}{5}$

$\rm\implies\Bigg ( \dfrac{5}{3} \Bigg )^x= \Bigg (\dfrac{5}{3}\Bigg )^{-1}$

Comparing the powers, we get,

$\boxed{\bf x=-1}$

Hope it helps...