Can $\log_{10}{x}, \ln(x), x$ form a geometric sequence? Does there exists a unique value of $x$, if any, such that $\log_{10}{x}, \ln(x), x$ forms a geometric sequence in some order?
This also might seem like a very odd question, since I can't see an immediate relationship between raising $10$ to something to get $x$ and doing the same but with $e$ to get $x$.

Using Ross's comment, we want a real solution to $\ln(x)/x=1/\ln(10).$ The right side is greater than $0.4,$ while the max over positive reals of the left side occurs at $x=e$ and is $1/e=0.3678\cdots$ so no real solution.
