What is the answer to 8+0.04x=6(1.02)^x ?

The answer to 8+0.04x=6(1.02)^x is

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8+0.04x=6(1.02)^x

8 + 0.04 x = 6 (1.02)^{x}

x = - \frac{0.04 W _{- 1} ( - \frac{6 l n ( 1.02 )}{0.04 e ^{\frac{8 l n ( 1.02 )}{0.04}}} ) + 8 ln ( 1.02 )}{0.04 ln ( 1.02 )}, x = - \frac{0.04 W _{0} ( - \frac{6 l n ( 1.02 )}{0.04 e ^{\frac{8 l n ( 1.02 )}{0.04}}} ) + 8 ln ( 1.02 )}{0.04 ln ( 1.02 )}

Decimal Notation

x = 19.14349 \dots, x = - 196.96500 \dots

Solution steps

8 + 0.04 x = 6 (1.02)^{x}

Prepare

8 + 0.04 x = 6 (1.02)^{x}

for Lambert form:

(8 + 0.04 x) e^{- l n (1.02) x} = 6

Rewrite the equation with

(- x - \frac{8}{0.04}) ln (1.02) = u

and

x = - \frac{0.04 u + 8 ln ( 1.02 )}{0.04 ln ( 1.02 )}

(8 + 0.04 (- \frac{0.04 u + 8 ln ( 1.02 )}{0.04 ln ( 1.02 )})) e^{- l n (1.02) (- \frac{0.04 u + 8 l n ( 1.02 )}{0.04 l n ( 1.02 )})} = 6

Simplify

e^{\frac{0.04 u + 8 l n ( 1.02 )}{0.04}} (8 - \frac{0.04 u + 8 ln ( 1.02 )}{ln ( 1.02 )}) = 6

Rewrite

e^{\frac{0.04 u + 8 l n ( 1.02 )}{0.04}} (8 - \frac{0.04 u + 8 ln ( 1.02 )}{ln ( 1.02 )}) = 6

in Lambert form:

\frac{e ^{\frac{0.04 u + 8 l n ( 1.02 )}{0.04}} u}{e ^{\frac{8 l n ( 1.02 )}{0.04}}} = - \frac{6 ln ( 1.02 )}{0.04 e ^{\frac{8 l n ( 1.02 )}{0.04}}}

Solve

\frac{e ^{\frac{0.04 u + 8 l n ( 1.02 )}{0.04}} u}{e ^{\frac{8 l n ( 1.02 )}{0.04}}} = - \frac{6 ln ( 1.02 )}{0.04 e ^{\frac{8 l n ( 1.02 )}{0.04}}} : u = W_{- 1} (- \frac{6 ln ( 1.02 )}{0.04 e ^{\frac{8 l n ( 1.02 )}{0.04}}}), u = W_{0} (- \frac{6 ln ( 1.02 )}{0.04 e ^{\frac{8 l n ( 1.02 )}{0.04}}})

u = W_{- 1} (- \frac{6 ln ( 1.02 )}{0.04 e ^{\frac{8 l n ( 1.02 )}{0.04}}}), u = W_{0} (- \frac{6 ln ( 1.02 )}{0.04 e ^{\frac{8 l n ( 1.02 )}{0.04}}})

Substitute back

u = (- x - \frac{8}{0.04}) ln (1.02),

solve for

x

Solve

(- x - \frac{8}{0.04}) ln (1.02) = W_{- 1} (- \frac{6 ln ( 1.02 )}{0.04 e ^{\frac{8 l n ( 1.02 )}{0.04}}}) : x = - \frac{0.04 W _{- 1} ( - \frac{6 l n ( 1.02 )}{0.04 e ^{\frac{8 l n ( 1.02 )}{0.04}}} ) + 8 ln ( 1.02 )}{0.04 ln ( 1.02 )}

Solve

(- x - \frac{8}{0.04}) ln (1.02) = W_{0} (- \frac{6 ln ( 1.02 )}{0.04 e ^{\frac{8 l n ( 1.02 )}{0.04}}}) : x = - \frac{0.04 W _{0} ( - \frac{6 l n ( 1.02 )}{0.04 e ^{\frac{8 l n ( 1.02 )}{0.04}}} ) + 8 ln ( 1.02 )}{0.04 ln ( 1.02 )}

x = - \frac{0.04 W _{- 1} ( - \frac{6 l n ( 1.02 )}{0.04 e ^{\frac{8 l n ( 1.02 )}{0.04}}} ) + 8 ln ( 1.02 )}{0.04 ln ( 1.02 )}, x = - \frac{0.04 W _{0} ( - \frac{6 l n ( 1.02 )}{0.04 e ^{\frac{8 l n ( 1.02 )}{0.04}}} ) + 8 ln ( 1.02 )}{0.04 ln ( 1.02 )}

x^{10} = e^{x}

4^{x} = \frac{8}{1 6 ^{x}}

x, e^{x} = 4

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

e^{x - 1} = 3^{\frac{2 x}{3}}