Question

What is the exact solution for the equation

Answer 1

Option C is correct.

Exact solution of
`2^(x+2) = 5^(2x)` is,
`(2\ln 2)/(2\ln 5 - \ln 2)`

Explanation:

Given the equation:
`2^(x+2) = 5^(2x)`

Using logarithmic rules:

• 
  `a^x=b^y` ⇒
  `x \ln a =y \ln b`

• 
  `\ln x^a = a\ln x`

Given:
`2^(x+2) = 5^(2x)`

Taking logarithmic both sides:

`\ln 2^(x+2) = \ln 5^(2x)`

By logarithmic rules;

`(x+2) \ln 2 = (2x) \ln 5`

Using distributive property i.e
`a\cdot (b+c) = a\cdot b + a\cdot c`

`x \ln 2 +2 \ln 2 = 2x \ln 5`

Subtract
`x \ln 2` from both sides we get;

`x \ln 2 +2 \ln 2 - x \ln 2= 2x \ln 5 - x \ln 2`

Simplify:

`2 \ln 2= 2x \ln 5 - x \ln 2`

or

`2 \ln 2= x (2\ln 5 - \ln 2)`

Divide both sides by
`(2\ln 5 - \ln 2)` we get;

`(2\ln 2)/(2\ln 5 - \ln 2) = x`

Therefore, the exact solution of the given equation is
`(2\ln 2)/(2\ln 5 - \ln 2)`