The caret ^ symbol means exponentiation informally in math. Why not a symbol for log too? Plus, minus, multiply, divide, and exponentiation all have symbols in math (+, -, *, /, ^ ) . But why isn't there the missing log symbol too? Here's how it would work:
4 ^ 5 = 1024 (as is standard for exponentiation)
1024 _ 4 = 5 ("_" is the new log operator!)
Look how much more elegant <1> is compared to <2>, <3> or <4>. We shouldn't need to do those 'hacks' to express the same thing:
1: 1024 _ 4 = 5
2: log(1024)/log(4) = 5
3: LogBase(1024,4) = 5
4: log4(1024) = 5
NB: It doesn't have to be an underscore symbol. It's just the first thing that sprang to mind.
Having a binary log operator would be useful for visually parsing the sum due to its conciseness. Additionally, using root symbols (for exponentiation's other inverse) eats up vertical space, and I think there's value in being able to express a sum on a single line. It's also easier to copy and paste a single line for use elsewhere when we use standard text symbols that are available on a keyboard.
 A: ^ is not really a mathematical symbol for exponentiation. The mathematical notation is to write the power as a superscript. ^ is just how the superscript is commonly (but not always) represented on computers in context where one has to stick to linear sequences of characters.
In any case, the answer is that there are many different functions that one might want a compact notation for, and only so many different symbols that one can reasonably expect people to remember. At some point using names becomes easier -- for example a name made up of letters can easily be looked up in an alphabetically arranged index; that is much harder if we just use some strange graphical symbol that doesn't have a conventional place in the alphabet.
Having a specialized notation for exponentiation is worthwhile because exponentiation is so common. In particular we use it to write down polynomials, which are very important functions.
Logarithms don't nearly reach the same level of importance among all the other functions one might also want specialized notations for.
A: All of the other examples (+,-,*,/,^) are binary operations.  
Log is generally a unary operation.  Either we are taking all of our logs base e, or base 10, or base 2.  Only very rarely do we need various bases in a single passage.  The proposed notation commits you to a binary notation, where the base of the log must be repeated every time.  log(x) instead omits that base, once it has been specified.  Hence the popularity of $\ln x$, $\log x$, $\lg x$ to denote these three bases respectively (although sometimes we use the middle one for something else).
A: 
But why isn't there the missing log symbol too?

The answer, as many people have pointed out, is "historical accident." The evolution of language is a rich, messy process whose details are hard to predict, control, or explain. For a glimpse of how our current mathematical vocabulary came to be, check out Florian Cajori's History of Mathematical Notations. Since Cajori died in 1930, his book is currently out of copyright in  the U.S., so depending on where you live, you may be able to read and distribute it freely.
Notation for logarithms is discussed in the second volume of Cajori's book, paragraph 469. Intriguingly, Cajori doesn't document any notations like yours, with the logarithm appearing as a binary operator between the argument and the base. This may be because when people use logarithms, the base is often fixed, so it's inconvenient to mention the base at all.
A similar thing happens with exponents. In many situations (including exponentiation in Lie groups more complicated than the positive reals), it's easiest to work exclusively in (the generalization of) base $e$, and just write the exponential of $x$ as $\exp x$

I quite like your notation (although I'd strongly prefer a symbol other than _, since many people already use that for subscripts), and I hope it catches on! I'll even suggest a modification. If you change the order of the arguments so the base comes first, just like with powers, exponentiation and logarithms cancel neatly:
b ^ (b L x) = x
b L (b ^ x) = x
A: As Henning Makholm notes, ^ is much more a "computer notation" rather than mathematical notation. And so is _. If you want to be strict about a "duality" between the mathematical notation of exponentiation as superscript, for logarithm you'd have to use subscript (which actually coincides with _ in LaTeX). So the inverse of $e^x$ (i.e. the natural logarihtm) would be $e_x$ in this "perfectly dual" notation. The obvious trouble with using subscript for logarithm is that there are great many other places/cases where subscripts are useful. So there would be a lot of confusion. Occasionally, but less often, superscripts too are used for denoting things other than exponentiation.
