Question

5^log x+3^log x=3^log x+1-5^log x-1

Answer 1

Please see the attachment

Answer 2

Answer:

x = 10

Step-by-step explanation:

The given equation is

$5^{\log x}+3^{\log x}=3^{\log x+1}-5^{\log x-1}$

Let as assume log x = t. So, the given equation can be written as

$5^{t}+3^{t}=3^{t+1}-5^{t-1}$

$5^{t}+5^{t-1}=3^{t+1}-3^{t}$

Using product and quotient property of exponent.

$5^{t}+\frac{5^{t}}{5}=3^{t}\cdot 3-3^{t}$     $[\because a^{m+n}=a^ma^n,\frac{a^m}{a^n}=a^{m-n}]$

$5\cdot 5^{t}+\frac{5^{t}}{5}=3^{t}\cdot 3-3^{t}$

$5^{t}(1+\frac{1}{5})=3^{t}(3-1)$

$5^{t}(\frac{6}{5})=3^{t}(2)$

Isolate the variable terms on one side.

$\frac{5^t}{3^t}=\frac{5}{6}\times 2$

$(\frac{5}{3})^t=\frac{5}{3}$

It can be written as

$(\frac{5}{3})^t=(\frac{5}{3})^1$

On comparing both sides we get

$t=1$

$log x= 1$          $[\because t=\log x]$

$log x= log 10$                $[\because \log 10=1]$

$x=10$

Therefore, the value of x is 10.