Solve for x ? 2e^2x=4

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Answer 1 · 2018-02-20T04:55:08

$x = \frac{ln (2)}{2}$ $x=ln(2)/2$

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

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

Put the natural log function around both sides:

$ln (e^{2 x}) = ln (2)$ $ln( e^(2x))=ln(2)$

Recall the exponential logarithm property which states that

$log (a^{x}) = x log (a)$ $log( a^x)=xlog(a)$

and apply it here:

$2 x ln (e) = ln (2)$ $2xln(e)=ln(2)$

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

$ln (e) = {log}_{e} e = 1$ $ln(e)= log_e e=1$

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

So

$2 x = ln (2)$ $2x=ln(2)$

$x = \frac{ln (2)}{2}$ $x=ln(2)/2$

Answer 2 · 2018-02-20T07:24:46

$x = \frac{ln 2}{2} \approx 0.35$ $x=ln2/2~~0.35$

I'm assuming that your problem is

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

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

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

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

$ln (e^{2 x}) = ln 2$ $ln(e^(2x))=ln2$

Let $y = 2 x$ $y=2x$

$\Leftrightarrow ln (e^{y}) = ln 2$ $<=>ln(e^y)=ln2$

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

$:.y=ln2$

Substituting back $y=2x$ , we get

$2x=ln2$

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

$x=ln2/2$

Using a calculator, this yields $0.34657...~~0.35$