Smeagol's Genius Game

$\log_{3}{5}=1.465 \implies 3^{1.465}=5$

$(3^{1.465})^{360}=5^{360}$

$3^{527.4}=5^{360}$

Hence, the smallest $b$ is $\boxed {528}$

Hello,

as given that creature smeagol knew that log 5 base 3 = 1.465,

so,

3^(b) > 5 ^ 360

3^(b) = 5^(360)

b log 3 = 360 log 5, make the equation to base 3,

b log 3 base 3 = 360 log 5 base 3,

b(1) = 360 (1.465)

b = 527.4, remember that the condition,

3^(b) > 5^(360),

therefore b > 527.4 = 528 (the closest to get the value)...

thanks...

My computer program for this: long double a, b; a = pow(5, 360); for (b = 0; b < 1000; b++) { if (pow(3, b)>a) { cout << b<<endl; break; } }

3^b > 5^360,
b log 3 > 360 log 5,
b > 360*1.465,
b = 528.

log 5 to base 3=1.465

so 3^1.465=5

now express 3^b>5^360 as power of b

so 3^b>(3^1.465)^360

so 3^b>3^527.4

so nearest integer power greater than 527.24 is 528

so the answer is 528

360*1.465=527.4 So, the smallest integer greater than 527.4 is 528.