What is the answer to 4^x(5^2x)=10 ?

The answer to 4^x(5^2x)=10 is x=(\W_{0}((4ln(2))/5))/(2ln(2))

Solve for x: 4^x(5^2x)=10

The solution is x=(\W_{0}((4ln(2))/5))/(2ln(2))

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4^{x} (5^{2} x) = 10

x = \frac{W _{0} ( \frac{4 l n ( 2 )}{5} )}{2 ln ( 2 )}

Decimal Notation

x = 0.27370 \dots

Solution steps

4^{x} (5^{2} x) = 10

Prepare

4^{x} (5^{2} x) = 10

for Lambert form:

e^{2 l n (2) (x + 2 l o g_{4} (5))} x = 10

Rewrite the equation with

2 ln (2) x = u

and

x = \frac{u}{2 ln ( 2 )}

e^{2 l n (2) ((\frac{u}{2 l n ( 2 )}) + 2 l o g_{4} (5))} (\frac{u}{2 ln ( 2 )}) = 10

Simplify

\frac{e ^{u + 4 l n (2) l o g_{4} (5)} u}{2 ln ( 2 )} = 10

Rewrite

\frac{e ^{u + 4 l n (2) l o g_{4} (5)} u}{2 ln ( 2 )} = 10

in Lambert form:

e^{u} u = \frac{4 ln ( 2 )}{5}

Solve

e^{u} u = \frac{4 ln ( 2 )}{5} : u = W_{0} (\frac{4 ln ( 2 )}{5})

u = W_{0} (\frac{4 ln ( 2 )}{5})

Substitute back

u = 2 ln (2) x,

solve for

x

Solve

2 ln (2) x = W_{0} (\frac{4 ln ( 2 )}{5}) : x = \frac{W _{0} ( \frac{4 l n ( 2 )}{5} )}{2 ln ( 2 )}

x = \frac{W _{0} ( \frac{4 l n ( 2 )}{5} )}{2 ln ( 2 )}

40000 = 5400 \cdot 1.03 5^{x}

e^{9 x} = 4

2^{3 x} - 2^{7 x - 2} = 0

- 5 \cdot 7^{- 3 x - 6} - 8 = - 9

4 = 3^{x - 3}

What is the answer to 4^x(5^2x)=10 ?
The answer to 4^x(5^2x)=10 is x=(\W_{0}((4ln(2))/5))/(2ln(2))
Solve for x: 4^x(5^2x)=10
The solution is x=(\W_{0}((4ln(2))/5))/(2ln(2))