What is the answer to 5+0.5x=2(1.15)^x ?

The answer to 5+0.5x=2(1.15)^x is

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5+0.5x=2(1.15)^x

5 + 0.5 x = 2 (1.15)^{x}

x = - \frac{0.5 W _{- 1} ( - \frac{2 l n ( 1.15 )}{0.5 e ^{\frac{5 l n ( 1.15 )}{0.5}}} ) + 5 ln ( 1.15 )}{0.5 ln ( 1.15 )}, x = - \frac{0.5 W _{0} ( - \frac{2 l n ( 1.15 )}{0.5 e ^{\frac{5 l n ( 1.15 )}{0.5}}} ) + 5 ln ( 1.15 )}{0.5 ln ( 1.15 )}

Decimal Notation

x = 12.29178 \dots, x = - 8.83670 \dots

Solution steps

5 + 0.5 x = 2 (1.15)^{x}

Prepare

5 + 0.5 x = 2 (1.15)^{x}

for Lambert form:

(5 + 0.5 x) e^{- l n (1.15) x} = 2

Rewrite the equation with

(- x - \frac{5}{0.5}) ln (1.15) = u

and

x = - \frac{0.5 u + 5 ln ( 1.15 )}{0.5 ln ( 1.15 )}

(5 + 0.5 (- \frac{0.5 u + 5 ln ( 1.15 )}{0.5 ln ( 1.15 )})) e^{- l n (1.15) (- \frac{0.5 u + 5 l n ( 1.15 )}{0.5 l n ( 1.15 )})} = 2

Simplify

e^{\frac{0.5 u + 5 l n ( 1.15 )}{0.5}} (5 - \frac{0.5 u + 5 ln ( 1.15 )}{ln ( 1.15 )}) = 2

Rewrite

e^{\frac{0.5 u + 5 l n ( 1.15 )}{0.5}} (5 - \frac{0.5 u + 5 ln ( 1.15 )}{ln ( 1.15 )}) = 2

in Lambert form:

\frac{e ^{\frac{0.5 u + 5 l n ( 1.15 )}{0.5}} u}{e ^{\frac{5 l n ( 1.15 )}{0.5}}} = - \frac{2 ln ( 1.15 )}{0.5 e ^{\frac{5 l n ( 1.15 )}{0.5}}}

Solve

\frac{e ^{\frac{0.5 u + 5 l n ( 1.15 )}{0.5}} u}{e ^{\frac{5 l n ( 1.15 )}{0.5}}} = - \frac{2 ln ( 1.15 )}{0.5 e ^{\frac{5 l n ( 1.15 )}{0.5}}} : u = W_{- 1} (- \frac{2 ln ( 1.15 )}{0.5 e ^{\frac{5 l n ( 1.15 )}{0.5}}}), u = W_{0} (- \frac{2 ln ( 1.15 )}{0.5 e ^{\frac{5 l n ( 1.15 )}{0.5}}})

u = W_{- 1} (- \frac{2 ln ( 1.15 )}{0.5 e ^{\frac{5 l n ( 1.15 )}{0.5}}}), u = W_{0} (- \frac{2 ln ( 1.15 )}{0.5 e ^{\frac{5 l n ( 1.15 )}{0.5}}})

Substitute back

u = (- x - \frac{5}{0.5}) ln (1.15),

solve for

x

Solve

(- x - \frac{5}{0.5}) ln (1.15) = W_{- 1} (- \frac{2 ln ( 1.15 )}{0.5 e ^{\frac{5 l n ( 1.15 )}{0.5}}}) : x = - \frac{0.5 W _{- 1} ( - \frac{2 l n ( 1.15 )}{0.5 e ^{\frac{5 l n ( 1.15 )}{0.5}}} ) + 5 ln ( 1.15 )}{0.5 ln ( 1.15 )}

Solve

(- x - \frac{5}{0.5}) ln (1.15) = W_{0} (- \frac{2 ln ( 1.15 )}{0.5 e ^{\frac{5 l n ( 1.15 )}{0.5}}}) : x = - \frac{0.5 W _{0} ( - \frac{2 l n ( 1.15 )}{0.5 e ^{\frac{5 l n ( 1.15 )}{0.5}}} ) + 5 ln ( 1.15 )}{0.5 ln ( 1.15 )}

x = - \frac{0.5 W _{- 1} ( - \frac{2 l n ( 1.15 )}{0.5 e ^{\frac{5 l n ( 1.15 )}{0.5}}} ) + 5 ln ( 1.15 )}{0.5 ln ( 1.15 )}, x = - \frac{0.5 W _{0} ( - \frac{2 l n ( 1.15 )}{0.5 e ^{\frac{5 l n ( 1.15 )}{0.5}}} ) + 5 ln ( 1.15 )}{0.5 ln ( 1.15 )}

\frac{400}{( 1 + e ^{- x} )} = 350

3^{2 x - 1} = 5^{x + 1}

9^{x + 3} = 2 7^{x - 1}

- 2 \cdot (- 2)^{n - 1} = (- 2)^{n}

(5^{x - 3}) (3^{2 x - 8}) = 225