Let k, w, and z be positive constants. Which of the following is equivalent to log10(kz/w²)?

Steps to Solve

1. Apply Logarithmic Properties

Start with the expression to simplify:

$$ \log_{10}\left(\frac{kz}{w^2}\right) $$

Using the quotient rule of logarithms:

$$ \log_{10}\left(\frac{A}{B}\right) = \log_{10}(A) - \log_{10}(B) $$

We can rewrite it as:

$$ \log_{10}(kz) - \log_{10}(w^2) $$

2. Further Simplification

Now apply the product rule to the first part and the power rule to the second part:

• Product rule:

$$ \log_{10}(AB) = \log_{10}(A) + \log_{10}(B) $$

• Power rule:

$$ \log_{10}(A^n) = n \log_{10}(A) $$

Using these, we have:

$$ \log_{10}(k) + \log_{10}(z) - \log_{10}(w^2) $$

This becomes:

$$ \log_{10}(k) + \log_{10}(z) - 2\log_{10}(w) $$

3. Combine the Expression

Now, combine all the parts together:

$$ \log_{10}(k) + \log_{10}(z) - 2 \log_{10}(w) $$

This expression is equivalent to:

$$ \log_{10}(k) + \log_{10}(z) - \log_{10}(w^2) $$

Thus, our simplified expression matches one of the given options.