'Equations of Magnitude: Unleashing Olympic Exponential Power'

Welcome to a world where exponential equations and Olympic glory collide! In this exciting series, we'll explore the fascinating relationship between mathematics and the Olympic Games. From exponential growth to decay, we'll examine how mathematical models can predict and explain extraordinary athletic performances. Join us as we unleash the power of exponential equations and witness the magic of the Olympics come to life!
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[TheVoitel]

Easy:

6^(X-6) - 6^(Y-6) = 1295

Substitute: X, Y ← X-6, Y-6

6^X - 6^Y = 1295
[→ X > Y]
6 does not divide 1295 → Y <= 0

→ 6^(X - Y) - 1 = 1295 6^{-Y}
→ Y = 0 (as 6 does not divide 6^(X - Y) - 1)

→ 6^X - 1 = 1295 → 6^X = 1296 = 6^4 → X = 4

Resubstitute: X, Y ← X + 6, Y + 6
X = 10, Y = 6

The two lines arguing that Y = 0 can alternatively be done simply using the size of 1295:
As -6^Y < 0 and 6^X <=1 if X <= 0 we need X > 0. But then 6^X is an integer, so 6^X - 6^Y can only be an integer if 6^Y is an integer, so Y >= 0.

[MichaelPalm-jm]

Oh, come on!! The only power of 6 that is odd is 0, so Y must be 6. Find what power of 6 1296 is (4) and add 6 to get X.