The function \(\mathrm{u(t) = 100(8)^{t/9}}\) gives the number of users of a social media app t days after its launch....

1. TRANSLATE the problem information

• Given: \(\mathrm{u(t) = 100(8)^{(t/9)}}\) represents users after t days
• Find: Number of days for users to double from initial value
• TRANSLATE "double from initial value" means \(\mathrm{u(t) = 2 × u(0)}\)

2. INFER what we need first

• We need the initial value \(\mathrm{u(0)}\) before we can find when it doubles
• Initial value means \(\mathrm{t = 0}\)

3. Calculate the initial value

• \(\mathrm{u(0) = 100(8)^{(0/9)}}\)
• \(\mathrm{= 100(8)^0}\)
• \(\mathrm{= 100(1)}\)
• \(\mathrm{= 100}\) users

4. Set up the doubling equation

• For doubling: \(\mathrm{u(t) = 2 × 100 = 200}\)
• So: \(\mathrm{100(8)^{(t/9)} = 200}\)
• SIMPLIFY by dividing both sides by 100: \(\mathrm{8^{(t/9)} = 2}\)

5. INFER the solution strategy

• We have an exponential equation with different bases (8 and 2)
• Strategy: Convert both sides to the same base to use the property that if \(\mathrm{a^x = a^y}\), then \(\mathrm{x = y}\)

6. SIMPLIFY using base conversion

• Since \(\mathrm{8 = 2^3}\), we can write: \(\mathrm{(2^3)^{(t/9)} = 2^1}\)
• Using exponent rules: \(\mathrm{2^{(3t/9)} = 2^1}\)
• Since bases are equal: \(\mathrm{3t/9 = 1}\)

7. SIMPLIFY to solve for t

• \(\mathrm{3t/9 = 1}\)
• \(\mathrm{t/3 = 1}\)
• \(\mathrm{t = 3}\)

Answer: 3 days

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret "double from initial value" and try to solve \(\mathrm{8^{(t/9)} = 2}\) directly without first finding what the initial value actually is.

They might think doubling means the function equals 2, leading to the equation \(\mathrm{100(8)^{(t/9)} = 2}\), which gives a completely different (and much smaller) value for t. This leads to confusion and an incorrect numerical answer.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up \(\mathrm{8^{(t/9)} = 2}\) but struggle with the base conversion or make algebraic errors in the final steps.

They might not recognize that \(\mathrm{8 = 2^3}\), or make calculation errors when simplifying \(\mathrm{3t/9 = 1}\) to get \(\mathrm{t = 3}\). This leads to getting stuck partway through and potentially guessing among answer choices.

The Bottom Line:

This problem requires careful interpretation of "doubling" language and systematic algebraic manipulation of exponential equations. Success depends on methodically finding the baseline first, then strategically converting to a common base.