The question is asking to find the value of $\log_{10} \frac{10}{x}$, given that $\log_{10} x = k$.

Solution:

We start with the given expression: $\log_{10} \frac{10}{x}$ Using the logarithmic property: $\log_a \frac{b}{c} = \log_a b - \log_a c$ we can rewrite: $\log_{10} \frac{10}{x} = \log_{10} 10 - \log_{10} x$

Now, substitute values:

1. We know $\log_{10} 10 = 1$ (logarithm of 10 in base 10 is 1).
2. From the problem, $\log_{10} x = k$.

Thus: $\log_{10} \frac{10}{x} = 1 - k$

Final Answer:

$\log_{10} \frac{10}{x} = 1 - k$

Would you like further explanation or details? Here are some related questions for practice:

1. If $\log_{10} x = 2$, what is $\log_{10} \frac{100}{x}$?
2. Simplify $\log_{10} (10x)$ in terms of $k$.
3. Prove the logarithmic property $\log_a \frac{b}{c} = \log_a b - \log_a c$.
4. If $\log_{10} x = k$, what is $\log_{10} \frac{x}{10}$?
5. What is the value of $\log_{10} x^2$ if $\log_{10} x = k$?

Tip: Always remember the basic logarithmic properties to simplify and manipulate logarithmic expressions efficiently!