What is the answer to 5^{-3}*x+5=0.04^x-3 ?

The answer to 5^{-3}*x+5=0.04^x-3 is x=(\W(250ln(5)*5^{2000}))/(2ln(5))-1000

Solve for x: 5^{-3}*x+5=0.04^x-3

The solution is x=(\W(250ln(5)*5^{2000}))/(2ln(5))-1000

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5^{- 3} \cdot x + 5 = 0.0 4^{x} - 3

x = \frac{W ( 250 ln ( 5 ) \cdot 5 ^{2000} )}{2 ln ( 5 )} - 1000

Solution steps

5^{- 3} x + 5 = 0.0 4^{x} - 3

Prepare

5^{- 3} \cdot x + 5 = 0.0 4^{x} - 3

for Lambert form:

1 = e^{2 l n (5) x} (\frac{1}{125} x + 8)

Rewrite the equation with

2 ln (5) (x + 1000) = u

and

x = \frac{u}{2 ln ( 5 )} - 1000

1 = e^{2 l n (5) (\frac{u}{2 l n ( 5 )} - 1000)} (\frac{1}{125} (\frac{u}{2 ln ( 5 )} - 1000) + 8)

Simplify

1 = \frac{e ^{u - 2000 l n (5)} u}{250 ln ( 5 )}

Rewrite

1 = \frac{e ^{u - 2000 l n (5)} u}{250 ln ( 5 )}

in Lambert form:

e^{u} u = 250 ln (5) \cdot 5^{2000}

Solve

e^{u} u = 250 ln (5) \cdot 5^{2000} : u = W (250 ln (5) \cdot 5^{2000})

u = W (250 ln (5) \cdot 5^{2000})

Substitute back

u = 2 ln (5) (x + 1000),

solve for

x

Solve

2 ln (5) (x + 1000) = W (250 ln (5) \cdot 5^{2000}) : x = \frac{W ( 250 ln ( 5 ) \cdot 5 ^{2000} )}{2 ln ( 5 )} - 1000

x = \frac{W ( 250 ln ( 5 ) \cdot 5 ^{2000} )}{2 ln ( 5 )} - 1000

2^{x} \cdot 0. 2^{5} = 2^{10}

2^{2 x + 1} - 2^{x + 4} = 2^{x + 2} - 32

2^{x + 4} - 2^{x + 2} = 1

4^{x} + 1 0^{x} = 2 5^{x}

\frac{1}{2 ^{x}} + \frac{1}{2 ^{x}} + \frac{1}{2 ^{x}} = 3

What is the answer to 5^{-3}*x+5=0.04^x-3 ?
The answer to 5^{-3}*x+5=0.04^x-3 is x=(\W(250ln(5)*5^{2000}))/(2ln(5))-1000
Solve for x: 5^{-3}*x+5=0.04^x-3
The solution is x=(\W(250ln(5)*5^{2000}))/(2ln(5))-1000