Question

What is the exact solution for the equation

Answer 1

Answer:

Option C is correct.

Exact solution of  $2^{x+2} = 5^{2x}$ is, $\frac{2\ln 2}{2\ln 5 - \ln 2}$

Step-by-step 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;

$\frac{2\ln 2}{2\ln 5 - \ln 2} = x$

Therefore, the exact solution of the given equation is $\frac{2\ln 2}{2\ln 5 - \ln 2}$