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

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

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(5x)^x=5^{5^5}

(5 x)^{x} = 5^{5^{5}}

x = 625

Solution steps

(5 x)^{x} = 5^{5^{5}}

Prepare

(5 x)^{x} = 5^{5^{5}}

for Lambert form:

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

Rewrite the equation with

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

and

x = \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}{x}) ln (5),

solve for

x

Solve

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

x = 625

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