Highest Exponent

TIP101 Unit 4 Session 1 (Click for link to problem statements)

U-nderstand

Understand what the interviewer is asking for by using test cases and questions about the problem.

• What if the base is 1?
  • Every power of 1 is still 1, so no finite highest exponent exists. The function should reject this input (e.g. raise a ValueError) rather than loop forever.
• What if the limit is less than the base?
  • The highest exponent should be 0, as any positive exponent would exceed the limit.

P-lan

Plan the solution with appropriate visualizations and pseudocode.

General Idea: Start with the smallest exponent and multiply the base until the result exceeds the limit.

1) If `base` is 1, raise an error — every power of 1 is 1, so no highest exponent exists.
2) Initialize `exponent` to 0, representing the smallest exponent.
3) Initialize `power` to 1, which is base^0.
4) While multiplying the current `power` by the base stays within the `limit`:
  a) Multiply `power` by `base` to get the next power.
  b) Increment `exponent` by 1 to reflect the next higher power level.
5) Once the loop exits, `exponent` equals the number of successful multiplications — the largest `exponent` with base^exponent <= limit — so return it.

⚠️ Common Mistakes

• Forgetting to handle edge cases where base is 1 or limit is less than base.
• Incorrectly updating the power or misplacing the increment of exponent.

I-mplement

def find_highest_exponent(base, limit):
    if base == 1:
        # Every power of 1 is 1, so no finite highest exponent exists
        raise ValueError("base 1 has no highest exponent: 1**k equals 1 for every k")
    exponent = 0  # Start with an exponent of 0
    power = 1  # The result of base^exponent
    while power * base <= limit:
        power *= base
        exponent += 1  # Increment the exponent each time the base is multiplied
    return exponent