how can you simplify this??

log10(10^.5)

log10 as in log base 10

By definition:

Log_10[10^(x)] = x

ok, but it is 10 to the 1/2 power, not x
how would you simplify it then?

That's then the special case of x = 1/2:

If for all x:

Log_10[10^x] = x,

then for x = 1/2 this implies:

Log_10[10^(1/2)] = 1/2

To simplify the expression log10(10^.5), you can use the fact that log10(10^x) = x. In this case, since 10^.5 equals the square root of 10, you can rewrite the expression as log10(sqrt(10)).

Since log10(10^x) = x, we can say that log10(sqrt(10)) = x. Now, we want to find the value of x that satisfies this equation.

To do that, we can recall that log10(sqrt(10)) can be rewritten as the exponent form 10^x = sqrt(10).

Now, we can solve for x by raising both sides of the equation to the power of 2 (so we can eliminate the square root):

(10^x)^2 = (sqrt(10))^2

10^(2x) = 10

Now, we can equate the exponents:

2x = 1

Divide both sides by 2:

x = 1/2

So, the simplified version of log10(10^.5) is 1/2.