Solve both parts and justify the conditions used. Part A: Find the base-10 logarithmic increase when a positive quantity is multiplied by 1000. Part B: Show that f(x)=2ˣ−3 has a zero between 1 and 2, and explain what else makes the zero unique.

1. Part A reasoning

2. Translate the context into an exponential or logarithmic equation and check what the model assumes.

3. Calculate or simplify this relation.

   $$\log_{10}(1000x/x)=\log_{10}1000=3$$

4. The result is independent of the initial positive x.

5. Part B reasoning

6. Translate the context into an exponential or logarithmic equation and check what the model assumes.

7. Calculate or simplify this relation.

   $$f(1)=-1<0,\quad f(2)=1>0$$

8. A sign change alone does not establish uniqueness.

A: The requested value is 3. B: Continuity gives existence; strict increase gives uniqueness.

Checks and common pitfalls: The result is independent of the initial positive x. A sign change alone does not establish uniqueness.