The function f is defined by \(\mathrm{f(x) = a \log_2(x + 4) + b}\), where a and b are constants....

1. TRANSLATE the point condition

• Given information:
  • \(\mathrm{f(x) = a \log_2(x + 4) + b}\)
  • Graph passes through \(\mathrm{(-3, 0)}\)
  • \(\mathrm{f(0) \gt 0}\)

• TRANSLATE 'passes through \(\mathrm{(-3, 0)}\)' means \(\mathrm{f(-3) = 0}\)

2. INFER the systematic approach

• We have two conditions that will help us determine information about a and b
• Start with the point condition since it gives us an exact equation

3. SIMPLIFY to find b

• Substitute the point condition:

\(\mathrm{f(-3) = 0}\)
\(\mathrm{a \log_2(-3 + 4) + b = 0}\)
\(\mathrm{a \log_2(1) + b = 0}\)

• Since \(\mathrm{\log_2(1) = 0}\):

\(\mathrm{a(0) + b = 0}\)
\(\mathrm{b = 0}\)

4. TRANSLATE and use the inequality condition

• Now use \(\mathrm{f(0) \gt 0}\):

\(\mathrm{f(0) = a \log_2(0 + 4) + b}\)
\(\mathrm{f(0) = a \log_2(4) + 0}\)

• Since \(\mathrm{\log_2(4) = 2}\):

\(\mathrm{f(0) = 2a}\)

5. APPLY CONSTRAINTS from the inequality

• From \(\mathrm{f(0) \gt 0}\):

\(\mathrm{2a \gt 0}\)
Therefore \(\mathrm{a \gt 0}\)

6. INFER which answer choice is correct

• We determined: \(\mathrm{a \gt 0}\) and \(\mathrm{b = 0}\)
• Check each choice:

1. \(\mathrm{a \lt 0}\): False
2. \(\mathrm{b \gt 0}\): False
3. \(\mathrm{a \lt b}\) means \(\mathrm{a \lt 0}\): False
4. \(\mathrm{a \gt b}\) means \(\mathrm{a \gt 0}\): True

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Not remembering that \(\mathrm{\log_2(1) = 0}\) or \(\mathrm{\log_2(4) = 2}\)

Students might try to evaluate these logarithms incorrectly, thinking \(\mathrm{\log_2(1) = 1}\) or being unsure about \(\mathrm{\log_2(4)}\). This leads to incorrect equations and makes it impossible to solve for the parameters accurately. This causes them to get stuck and guess.

Second Most Common Error:

Weak INFER skill: Not systematically using both given conditions

Some students might only use one condition (either the point or the inequality) and not realize they need both pieces of information to fully determine the relationship between a and b. They might jump to conclusions too quickly without working through both constraints. This may lead them to select Choice B \(\mathrm{(b \gt 0)}\) by incorrectly assuming b must be positive.

The Bottom Line:

This problem requires students to methodically work through multiple conditions involving logarithmic functions. Success depends on knowing basic logarithm values and systematically applying all given constraints.