# Solve for x ? 2e^2x=4

Feb 20, 2018

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

#### Explanation:

$2 {e}^{2 x} = 4$

${e}^{2 x} = 2$

Put the natural log function around both sides:

$\ln \left({e}^{2 x}\right) = \ln \left(2\right)$

Recall the exponential logarithm property which states that

$\log \left({a}^{x}\right) = x \log \left(a\right)$

and apply it here:

$2 x \ln \left(e\right) = \ln \left(2\right)$

$\ln \left(e\right) = 1$ , this is from the basic definition of the natural log function.

$\ln \left(e\right) = {\log}_{e} e = 1$

since ${e}^{1} = e$.

So

$2 x = \ln \left(2\right)$

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

Feb 20, 2018

$x = \ln \frac{2}{2} \approx 0.35$

#### Explanation:

I'm assuming that your problem is

$2 {e}^{2 x} = 4$

In this case, we first have to divide by $2$ to get

${e}^{2 x} = 2$

If we now take the natural log of both sides, we get

$\ln \left({e}^{2 x}\right) = \ln 2$

Let $y = 2 x$

$\iff \ln \left({e}^{y}\right) = \ln 2$

Recall that, $\ln \left({e}^{y}\right) = y$

$\therefore y = \ln 2$

Substituting back $y = 2 x$, we get

$2 x = \ln 2$

Now, we just have to divide by $2$ to isolate $x$:

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

Using a calculator, this yields $0.34657 \ldots \approx 0.35$