What is the answer to 10^1-x=6^x ?

The answer to 10^1-x=6^x is x=-(\W_{0}(60466176ln(6))-10ln(6))/(ln(6))

Solve for x: 10^1-x=6^x

The solution is x=-(\W_{0}(60466176ln(6))-10ln(6))/(ln(6))

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1 0^{1} - x = 6^{x}

x = - \frac{W _{0} ( 60466176 ln ( 6 ) ) - 10 ln ( 6 )}{ln ( 6 )}

Decimal Notation

x = 1.21293 \dots

Solution steps

1 0^{1} - x = 6^{x}

Prepare

1 0^{1} - x = 6^{x}

for Lambert form:

e^{- l n (6) x} (10 - x) = 1

Rewrite the equation with

(- x + 10) ln (6) = u

and

x = - \frac{u - 10 ln ( 6 )}{ln ( 6 )}

e^{- l n (6) (- \frac{u - 10 l n ( 6 )}{l n ( 6 )})} (10 - (- \frac{u - 10 ln ( 6 )}{ln ( 6 )})) = 1

Simplify

e^{u - 10 l n (6)} (10 + \frac{u - 10 ln ( 6 )}{ln ( 6 )}) = 1

Rewrite

e^{u - 10 l n (6)} (10 + \frac{u - 10 ln ( 6 )}{ln ( 6 )}) = 1

in Lambert form:

e^{u} u = 60466176 ln (6)

Solve

e^{u} u = 60466176 ln (6) : u = W_{0} (60466176 ln (6))

u = W_{0} (60466176 ln (6))

Substitute back

u = (- x + 10) ln (6),

solve for

x

Solve

(- x + 10) ln (6) = W_{0} (60466176 ln (6)) : x = - \frac{W _{0} ( 60466176 ln ( 6 ) ) - 10 ln ( 6 )}{ln ( 6 )}

x = - \frac{W _{0} ( 60466176 ln ( 6 ) ) - 10 ln ( 6 )}{ln ( 6 )}

1 1^{- 3 x} = 1 7^{x - 6}

2 (6^{x}) - 54 (2^{x}) - 16 (3^{x}) + 432 = 0

7^{x + 2} + 7^{x + 1} = 8

12000 = 200 e^{0.01 t}

4^{x} = 81

What is the answer to 10^1-x=6^x ?
The answer to 10^1-x=6^x is x=-(\W_{0}(60466176ln(6))-10ln(6))/(ln(6))
Solve for x: 10^1-x=6^x
The solution is x=-(\W_{0}(60466176ln(6))-10ln(6))/(ln(6))