Given log_10(x) + log_10(y) = z. Express x explicitly in terms of y and z.

Introduction / Context:
This problem checks your comfort with basic logarithm properties in base 10 (common logs). In particular, it uses the identity log_10(a) + log_10(b) = log_10(a*b). We are asked to isolate x in terms of y and z when the sum of two common logs equals z.

Given Data / Assumptions:

• log_10(x) + log_10(y) = z
• All quantities are positive reals with x > 0, y > 0 (required for logarithms).

Concept / Approach:
Combine the sum of logs into a single logarithm, then convert from logarithmic to exponential form to solve for x in terms of y and z.

Step-by-Step Solution:

Verification / Alternative check:
Substitute x = 10^z / y back: log_10(10^z / y) + log_10(y) = (log_10(10^z) − log_10(y)) + log_10(y) = z − log_10(y) + log_10(y) = z ✓.

Why Other Options Are Wrong:

• z / y and 10 / z ignore exponentiation implied by the log equation.
• y / 10^z is the reciprocal of the correct expression.

Common Pitfalls:
Forgetting that adding logs multiplies arguments, and that solving for a variable inside a log requires converting to exponent form.

Final Answer:
10^z / y