We are given that $\log_{10} a = x$, $\log_{10} b = y$, and $\log_{10} c = z$. We need to express $\log_{10} \sqrt{\frac{10a}{b^5 c}}$ in terms of $x$, $y$, and $z$.

1. Problem Description

We are given that $\log_{10} a = x$, $\log_{10} b = y$, and $\log_{10} c = z$. We need to express $\log_{10} \sqrt{\frac{10a}{b^5 c}}$ in terms of $x$, $y$, and $z$.

2. Solution Steps

First, rewrite the square root as a power of 1/2:

$\log_{10} \sqrt{\frac{10a}{b^5 c}} = \log_{10} \left(\frac{10a}{b^5 c}\right)^{1/2}$

Use the power rule of logarithms: $\log_b (x^p) = p \log_b x$

$\log_{10} \left(\frac{10a}{b^5 c}\right)^{1/2} = \frac{1}{2} \log_{10} \left(\frac{10a}{b^5 c}\right)$

Use the quotient rule of logarithms: $\log_b \frac{x}{y} = \log_b x - \log_b y$

$\frac{1}{2} \log_{10} \left(\frac{10a}{b^5 c}\right) = \frac{1}{2} \left[\log_{10}(10a) - \log_{10}(b^5 c)\right]$

Use the product rule of logarithms: $\log_b (xy) = \log_b x + \log_b y$

$\frac{1}{2} \left[\log_{10}(10a) - \log_{10}(b^5 c)\right] = \frac{1}{2} \left[\log_{10} 10 + \log_{10} a - (\log_{10} b^5 + \log_{10} c)\right]$

Use the power rule of logarithms again:

$\frac{1}{2} \left[\log_{10} 10 + \log_{10} a - (\log_{10} b^5 + \log_{10} c)\right] = \frac{1}{2} \left[\log_{10} 10 + \log_{10} a - (5\log_{10} b + \log_{10} c)\right]$

Since $\log_{10} 10 = 1$, we have:

$\frac{1}{2} \left[1 + \log_{10} a - (5\log_{10} b + \log_{10} c)\right]$

Now substitute the given values $\log_{10} a = x$, $\log_{10} b = y$, and $\log_{10} c = z$:

$\frac{1}{2} \left[1 + x - (5y + z)\right] = \frac{1}{2} \left[1 + x - 5y - z\right]$

3. Final Answer

$\frac{1 + x - 5y - z}{2}$