Question

What is the exact solution to the equation 2^x+2 =5^2x ? A) −2ln2/2ln5−ln2 B) −ln2/2ln5+ln2 C) ln2/2ln5+ln2 D) 2ln2/2ln5−ln2

Answer 1

Answer: D

Step-by-step explanation:

          2ˣ⁺² = 5²ˣ

      ln 2ˣ⁺² = ln 5²ˣ               apply ln to both sides

(x + 2) ln 2 = 2x ln 5            apply exponent to coefficient log rule

x ln 2 + 2 ln 2 = 2x ln 5       distribute (x + 2) on left side

2 ln 2 = 2x ln 5 - x ln 2        subtract (x ln 2) from both sides

2 ln 2 = x(2 ln 5 - ln 2)         factor out (x) from right side

$\frac{2ln2}{2ln5 - ln2}$ = x                             divide both sides by (2 ln 5 - ln 2)

Answer 2

We want to get the exact solution for the equation:

$2^{x + 2} = 5^{2x}$

The correct option is D, the solution is:

$\frac{2*ln(2)}{2*ln(5) - ln(2)} = x$

Here we must use the property:

$ln(a^n) = n*ln(a)$

So we can apply the natural logarithm to both sides of our equation to get:

$ln(2^{x + 2}) = ln(5^{2x})\\\\(x + 2)*ln(2) = 2*x*ln(5)$

Now we can solve this for x.

$2*ln(2) = 2*x*ln(5) - x*ln(2) = x*(2*ln(5) - ln(2))\\\\\frac{2*ln(2)}{2*ln(5) - ln(2)} = x$

So the correct option is D.

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