What is the answer to 3^n=9n ?

The answer to 3^n=9n is n=3,n=-(\W_{0}(-(ln(3))/9))/(ln(3))

The solution is n=3,n=-(\W_{0}(-(ln(3))/9))/(ln(3))

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

3^n=9n

3^{n} = 9 n

n = 3, n = - \frac{W _{0} ( - \frac{l n ( 3 )}{9} )}{ln ( 3 )}

Decimal Notation

n = 3, n = 0.12786 \dots

Solution steps

3^{n} = 9 n

Prepare

3^{n} = 9 n

for Lambert form:

1 = e^{l n (3) (2 - n)} n

Rewrite the equation with

- n ln (3) = u

and

n = - \frac{u}{ln ( 3 )}

1 = e^{l n (3) (2 - (- \frac{u}{l n ( 3 )}))} (- \frac{u}{ln ( 3 )})

Simplify

1 = - \frac{e ^{2 l n (3) + u} u}{ln ( 3 )}

Rewrite

1 = - \frac{e ^{2 l n (3) + u} u}{ln ( 3 )}

in Lambert form:

e^{u} u = - \frac{ln ( 3 )}{9}

Solve

e^{u} u = - \frac{ln ( 3 )}{9} : u = - 3 ln (3), u = W_{0} (- \frac{ln ( 3 )}{9})

u = - 3 ln (3), u = W_{0} (- \frac{ln ( 3 )}{9})

Substitute back

u = - n ln (3),

solve for

n

Solve

- n ln (3) = - 3 ln (3) : n = 3

Solve

- n ln (3) = W_{0} (- \frac{ln ( 3 )}{9}) : n = - \frac{W _{0} ( - \frac{l n ( 3 )}{9} )}{ln ( 3 )}

n = 3, n = - \frac{W _{0} ( - \frac{l n ( 3 )}{9} )}{ln ( 3 )}

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

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}

What is the answer to 3^n=9n ?
The answer to 3^n=9n is n=3,n=-(\W_{0}(-(ln(3))/9))/(ln(3))
Solve for n: 3^n=9n
The solution is n=3,n=-(\W_{0}(-(ln(3))/9))/(ln(3))