Isolating for x in $\log_x(3\sqrt{x}) = k$

Question

I'm having trouble isolating for $x$ in $\log_x(3\sqrt{x}) = k$

I've tried various things. Here is what I ended up with:

$x^{k - \frac{1}{2}} = 3$

I don't know how to proceed. I keep getting stuck. Can anyone help?

Answer 1

I would have written the power as its definition, ie $x^{k-\frac12}=e^{(k-\frac12)\ln(x)}$ and then take the neperian log on both sides, and get $(k-\frac12) \ln(x)=\ln(3)$. Once you're here, it's quite easy to isolate $x$...

But every logarithm seems to work, so it's up to you !

Answer 2

May be

$(k-\frac{1}{2})logx = log3$

and $x = 10^{\left(\dfrac{log(3)}{k-\frac{1}{2}}\right)}$