What is the answer to (5y)^y=5^{5^5} ?

The answer to (5y)^y=5^{5^5} is y=625

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(5y)^y=5^{5^5}

(5 y)^{y} = 5^{5^{5}}

y = 625

Solution steps

(5 y)^{y} = 5^{5^{5}}

Prepare

(5 y)^{y} = 5^{5^{5}}

for Lambert form:

e^{l n (5) (1 - \frac{3125}{y})} y = 1

Rewrite the equation with

- (1 - \frac{3125}{y}) ln (5) = u

and

y = \frac{3125 ln ( 5 )}{u + ln ( 5 )}

e^{- u} (\frac{3125 ln ( 5 )}{u + ln ( 5 )}) = 1

Solve

e^{- u} (\frac{3125 ln ( 5 )}{u + ln ( 5 )}) = 1 : u = 4 ln (5)

u = 4 ln (5)

Substitute back

u = - (1 - \frac{3125}{y}) ln (5),

solve for

y

Solve

- (1 - \frac{3125}{y}) ln (5) = 4 ln (5) : y = 625

y = 625

5^{2 x} = 625

e^{3 x} = 15

e^{3 x} = 19

2^{2 x + 2} - 3 \cdot 2^{x + 2} + 8 = 0

4^{2 x} - 20 (4^{x}) + 64 = 0