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Popular >

4sqrt(x+1)=2^{3x-2}

Solution

4 x + 1 = 2^{3 x - 2}

Solution

x = - \frac{W _{- 1} ( - \frac{3 l n ( 2 )}{8192} ) + 6 ln ( 2 )}{6 ln ( 2 )}, x = - \frac{W _{0} ( - \frac{3 l n ( 2 )}{8192} ) + 6 ln ( 2 )}{6 ln ( 2 )}

+1

Solution steps

4 x + 1 = 2^{3 x - 2}

Prepare

4 x + 1 = 2^{3 x - 2}

for Lambert form:

e^{l n (2) (- 6 x + 8)} (x + 1) = 1

Rewrite the equation with

(- 6 x - 6) ln (2) = u

and

x = - \frac{u + 6 ln ( 2 )}{6 ln ( 2 )}

e^{l n (2) (- 6 (- \frac{u + 6 l n ( 2 )}{6 l n ( 2 )}) + 8)} ((- \frac{u + 6 ln ( 2 )}{6 ln ( 2 )}) + 1) = 1

Simplify

e^{u + 14 l n (2)} (- \frac{u + 6 ln ( 2 )}{6 ln ( 2 )} + 1) = 1

Rewrite

e^{u + 14 l n (2)} (- \frac{u + 6 ln ( 2 )}{6 ln ( 2 )} + 1) = 1

in Lambert form:

e^{u} u = - \frac{3 ln ( 2 )}{8192}

Solve

e^{u} u = - \frac{3 ln ( 2 )}{8192} : u = W_{- 1} (- \frac{3 ln ( 2 )}{8192}), u = W_{0} (- \frac{3 ln ( 2 )}{8192})

u = W_{- 1} (- \frac{3 ln ( 2 )}{8192}), u = W_{0} (- \frac{3 ln ( 2 )}{8192})

Substitute back

u = (- 6 x - 6) ln (2),

solve for

x

Solve

(- 6 x - 6) ln (2) = W_{- 1} (- \frac{3 ln ( 2 )}{8192}) : x = - \frac{W _{- 1} ( - \frac{3 l n ( 2 )}{8192} ) + 6 ln ( 2 )}{6 ln ( 2 )}

Solve

(- 6 x - 6) ln (2) = W_{0} (- \frac{3 ln ( 2 )}{8192}) : x = - \frac{W _{0} ( - \frac{3 l n ( 2 )}{8192} ) + 6 ln ( 2 )}{6 ln ( 2 )}

x = - \frac{W _{- 1} ( - \frac{3 l n ( 2 )}{8192} ) + 6 ln ( 2 )}{6 ln ( 2 )}, x = - \frac{W _{0} ( - \frac{3 l n ( 2 )}{8192} ) + 6 ln ( 2 )}{6 ln ( 2 )}

Verify Solutions:

x = - \frac{W _{- 1} ( - \frac{3 l n ( 2 )}{8192} ) + 6 ln ( 2 )}{6 ln ( 2 )}

True

, x = - \frac{W _{0} ( - \frac{3 l n ( 2 )}{8192} ) + 6 ln ( 2 )}{6 ln ( 2 )}

True

The solutions are

x = - \frac{W _{- 1} ( - \frac{3 l n ( 2 )}{8192} ) + 6 ln ( 2 )}{6 ln ( 2 )}, x = - \frac{W _{0} ( - \frac{3 l n ( 2 )}{8192} ) + 6 ln ( 2 )}{6 ln ( 2 )}

Verify Solutions:

x = - \frac{W _{- 1} ( - \frac{3 l n ( 2 )}{8192} ) + 6 ln ( 2 )}{6 ln ( 2 )}

True

, x = - \frac{W _{0} ( - \frac{3 l n ( 2 )}{8192} ) + 6 ln ( 2 )}{6 ln ( 2 )}

True

The solutions are

x = - \frac{W _{- 1} ( - \frac{3 l n ( 2 )}{8192} ) + 6 ln ( 2 )}{6 ln ( 2 )}, x = - \frac{W _{0} ( - \frac{3 l n ( 2 )}{8192} ) + 6 ln ( 2 )}{6 ln ( 2 )}

Popular Examples

e^{-4k-10}= 67/6

e^{- 4 k - 10} = \frac{67}{6}

3^{x} = 5^{x + 2}

(a^3)^6=(a^2)^x

(a^{3})^{6} = (a^{2})^{x}

solvefor x,4^{x+1}+2^y=20 solve for

x, 4^{x + 1} + 2^{y} = 20

solvefor t,(1.045)^t=2 solve for

t, (1.045)^{t} = 2