Question

use properties of logarithms to expand the logarithmic expression as much as possible. evaluate logarithmic expressions without using a calculator if possible.
\log \sqrt{10000z}
\log \sqrt{10000z} = \square

Explanation:

Step1: Rewrite the square root as a power

The square root of a number can be written as that number raised to the power of $\frac{1}{2}$. So, $\sqrt{10000z} = (10000z)^{\frac{1}{2}}$. Then the logarithmic expression becomes $\log(10000z)^{\frac{1}{2}}$.

Step2: Apply the power rule of logarithms

The power rule of logarithms states that $\log(a^b)=b\log(a)$. Applying this rule, we get $\frac{1}{2}\log(10000z)$.

Step3: Apply the product rule of logarithms

The product rule of logarithms states that $\log(ab)=\log(a)+\log(b)$. Applying this rule to $\log(10000z)$, we get $\log(10000)+\log(z)$. So now our expression is $\frac{1}{2}(\log(10000)+\log(z))$.

Step4: Evaluate $\log(10000)$

We know that $\log_{10}(10^n)=n$ (assuming base 10 for the logarithm, which is common if not specified). Since $10000 = 10^4$, $\log(10000)=\log(10^4) = 4$.

Step5: Simplify the expression

Substituting $\log(10000)=4$ into our expression, we have $\frac{1}{2}(4+\log(z))=\frac{4}{2}+\frac{1}{2}\log(z)=2+\frac{1}{2}\log(z)$.

Answer:

$2 + \frac{1}{2}\log(z)$ (or equivalent forms, assuming base 10 logarithm; if natural logarithm, it would be $2+\frac{1}{2}\ln(z)$ but the problem likely uses base 10 as 10000 is a power of 10)