Which of the following expressions most closely achieves both of the goals from the previous question, when solving the following equation:

e3x−8=1

Hint: You are expected to know that value of e0 and ln(1).

Question

Which of the following expressions most closely achieves both of the goals from the previous question, when solving the following equation:

e3x−8=1

Hint: You are expected to know that value of e0 and ln(1).

BlackTom AI · Accepted Answer

The question asks which expression most closely achieves both goals from the prior question when solving e^(3x−8) = 1, with the hint that e^0 = 1 and ln(1) = 0.
Option 1: x = (ln(1) + 8) / 3. Since ln(1) = 0, this simplifies to x = 8/3. This matches the solution to the equation because taking natural log of both sides yields 3x − 8 = ln(1) = 0, which then gives x = 8/3. However, as written, it presents the result in a single fraction form that combines the ln(1) term with the constant 8 before division; conceptually it is equivalent to the standard solution, but it is slightly roundabout compared to a direct x = 8/3.
Option 2: x = 0. Substituting x = 0 into the original equation gives e^(−8) on the left, which is not equal to 1. Therefore, this value does not satisfy the equation, so this choice is incorrect.
Option 3: x = 8/3. This is the straightforward numerical solution to 3x − 8 = 0, derived by setting the exponent equal to 0 (since e^t = 1 occurs at t = 0). It directly reflects the core reasoning: the exponent must be zero, so x = 8/3. This option is correct and matches the standard result.
Option 4: 3x − 8 = ln(1). Since ln(1) = 0, this equation reduces to 3x − 8 = 0, which again yields x = 8/3. This expresses the intermediate step obtained by taking natural logs of both sides of e^(3x−8) = 1, so it is also a valid representation of the solution process.
In summary, Options 1, 3, and 4 each encode a correct pathway to x = 8/3, albeit with different forms or levels of intermediate steps. Option 2 clearly does not satisfy the equation and is incorrect.

MAT136 - Calculus 2 - Winter 2025 Week 1 Tutorial Replacement Quiz

Which of the following expressions most closely achieves both of the goals from the previous question, when solving the following equation: e3x−8=1 Hint: You are expected to know that value of e0 and ln(1).

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