What is the answer to 10^xx10^{-9}=10^{11} ?

The answer to 10^xx10^{-9}=10^{11} is x=(\W_{0}(1000000000*10^{11}ln(10)))/(ln(10))

Solve for x: 10^xx10^{-9}=10^{11}

The solution is x=(\W_{0}(1000000000*10^{11}ln(10)))/(ln(10))

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1 0^{x} x 1 0^{- 9} = 1 0^{11}

x = \frac{W _{0} ( 1000000000 \cdot 1 0 ^{11} ln ( 10 ) )}{ln ( 10 )}

Decimal Notation

x = 18.72751 \dots

Solution steps

1 0^{x} x \cdot 1 0^{- 9} = 1 0^{11}

Prepare

1 0^{x} x 1 0^{- 9} = 1 0^{11}

for Lambert form:

e^{l n (10) (x - 9)} x = 1 0^{11}

Rewrite the equation with

x ln (10) = u

and

x = \frac{u}{ln ( 10 )}

e^{l n (10) ((\frac{u}{l n ( 10 )}) - 9)} (\frac{u}{ln ( 10 )}) = 1 0^{11}

Simplify

\frac{e ^{u - 9 l n (10)} u}{ln ( 10 )} = 1 0^{11}

Rewrite

\frac{e ^{u - 9 l n (10)} u}{ln ( 10 )} = 1 0^{11}

in Lambert form:

e^{u} u = 1000000000 \cdot 1 0^{11} ln (10)

Solve

e^{u} u = 1000000000 \cdot 1 0^{11} ln (10) : u = W_{0} (1000000000 \cdot 1 0^{11} ln (10))

u = W_{0} (1000000000 \cdot 1 0^{11} ln (10))

Substitute back

u = x ln (10),

solve for

x

Solve

x ln (10) = W_{0} (1000000000 \cdot 1 0^{11} ln (10)) : x = \frac{W _{0} ( 1000000000 \cdot 1 0 ^{11} ln ( 10 ) )}{ln ( 10 )}

x = \frac{W _{0} ( 1000000000 \cdot 1 0 ^{11} ln ( 10 ) )}{ln ( 10 )}

2 (3)^{x} = 8

3 \cdot 4 e^{2 - 2 x} + 1 = 4

4^{x^{2}} = 2^{8 x - 8}

4^{2 x + 1} = 8^{1 - x}

3^{x} + 1 = 0

What is the answer to 10^xx10^{-9}=10^{11} ?
The answer to 10^xx10^{-9}=10^{11} is x=(\W_{0}(1000000000*10^{11}ln(10)))/(ln(10))
Solve for x: 10^xx10^{-9}=10^{11}
The solution is x=(\W_{0}(1000000000*10^{11}ln(10)))/(ln(10))