If (0.2)x = 2 and log10 2 = 0.3010, then what is the value of x to the nearest…

Concept:

For an equation of the form ax = b, taking log (base 10) of both sides gives x·log(a) = log(b), so x = log(b) / log(a). Also, log(p/q) = log(p) - log(q), and log10(10) = 1.

Step-by-step solution:

1. Given equation: (0.2)x = 2.

2. Take log10 of both sides: x·log10(0.2) = log10(2).

3. Express 0.2 as 2/10, so log10(0.2) = log10(2) - log10(10) = 0.3010 - 1 = -0.6990.

4. Substitute: x·(-0.6990) = 0.3010.

5. Solve for x: x = 0.3010 / (-0.6990) ≈ -0.4306.

6. Round to the nearest tenth: x ≈ -0.4.

Cross-check:

Substituting x = -0.4306 back: log10((0.2)-0.4306) = -0.4306 × log10(0.2) = -0.4306 × (-0.6990) = 0.3010, which matches log10(2) = 0.3010, confirming (0.2)-0.4306 ≈ 2.