The function g is defined by \(\mathrm{g(x) = 5\log_2(x) - 15}\). What is the x-intercept of the graph of \(\mathrm{y...

1. TRANSLATE the problem information

• We need to find the x-intercept of \(\mathrm{g(x) = 5log_2(x) - 15}\)
• X-intercept occurs where the graph crosses the x-axis, meaning \(\mathrm{y = 0}\)
• This means we need to solve: \(\mathrm{g(x) = 0}\)

2. TRANSLATE to set up the equation

• Setting \(\mathrm{g(x) = 0}\) gives us: \(\mathrm{5log_2(x) - 15 = 0}\)

3. SIMPLIFY to isolate the logarithm

• Add 15 to both sides: \(\mathrm{5log_2(x) = 15}\)
• Divide both sides by 5: \(\mathrm{log_2(x) = 3}\)

4. INFER the conversion strategy

• We have \(\mathrm{log_2(x) = 3}\), which means "2 to what power equals x?"
• To find x, we need to convert from logarithmic to exponential form

5. SIMPLIFY the exponential calculation

• If \(\mathrm{log_2(x) = 3}\), then \(\mathrm{x = 2^3 = 8}\)
• Therefore, the x-intercept is \(\mathrm{(8, 0)}\)

Answer: B \(\mathrm{(8, 0)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often don't realize that "finding the x-intercept" means solving \(\mathrm{g(x) = 0}\). Instead, they might try to find where \(\mathrm{x = 0}\), which would give them the y-intercept. Or they get confused about what an intercept actually represents and attempt unrelated approaches.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Missing conceptual knowledge about logarithmic/exponential relationship: After correctly reaching \(\mathrm{log_2(x) = 3}\), students don't know how to convert this to find x. They might think \(\mathrm{x = 3}\) directly, since that's the number on the right side of the equation.

This may lead them to select Choice A \(\mathrm{(3, 0)}\).

The Bottom Line:

This problem tests whether students understand what x-intercepts represent and can work with logarithmic functions. The key insight is recognizing that finding intercepts means setting the function equal to zero, then systematically solving the resulting equation.