91Ó°ÊÓ

The first step is to simplify the expression \(\frac{1}{\sqrt{10000}}\). The square root of 10000 is 100, as \(100^2 = 10000\). So, we can rewrite the expression as \(\frac{1}{100}\). #Step 2: Use the properties of logarithms#

Now that we have simplified the expression inside the logarithm, we can rewrite it as follows: \[ \log{\frac{1}{100}} \] We will use the properties of logarithms to further break down and evaluate the expression. To do so, we first need to recognize that \(\frac{1}{100} = 10^{-2}\). Therefore, we can rewrite the expression as: \[ \log{10^{-2}} \] According to the power property of logarithms, \(\log{a^b} = b\log{a}\). In this case, \(a = 10\) and \(b = -2\). Applying the power property, the expression can be rewritten as follows: \[ -2\log{10} \] Lastly, we know that \(\log{10} = 1\) since the logarithm with base 10 of 10 is equal to 1. This allows us to rewrite the expression as: \[ -2(1) \] #Step 3: Calculate the final result#

Multiplying -2 by 1 gives us the final result: \[ -2(1) = -2 \] So, the value of the given expression \(\log \frac{1}{\sqrt{10000}}\) is -2.