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Popular >

7^4x+1=5^x

Solution

7^{4} x + 1 = 5^{x}

Solution

x = - \frac{2401 W _{- 1} ( - \frac{l n ( 5 )}{2401 e ^{\frac{l n ( 5 )}{2401}}} ) + ln ( 5 )}{2401 ln ( 5 )}, x = - \frac{2401 W _{0} ( - \frac{l n ( 5 )}{2401 e ^{\frac{l n ( 5 )}{2401}}} ) + ln ( 5 )}{2401 ln ( 5 )}

+1

Decimal Notation

x = 5.94371 \dots, x = 0

Solution steps

7^{4} x + 1 = 5^{x}

Prepare

7^{4} x + 1 = 5^{x}

for Lambert form:

e^{- l n (5) x} (2401 x + 1) = 1

Rewrite the equation with

(- x - \frac{1}{2401}) ln (5) = u

and

x = - \frac{2401 u + ln ( 5 )}{2401 ln ( 5 )}

e^{- l n (5) (- \frac{2401 u + l n ( 5 )}{2401 l n ( 5 )})} (2401 (- \frac{2401 u + ln ( 5 )}{2401 ln ( 5 )}) + 1) = 1

Simplify

e^{\frac{2401 u + l n ( 5 )}{2401}} (- \frac{2401 u + ln ( 5 )}{ln ( 5 )} + 1) = 1

Rewrite

e^{\frac{2401 u + l n ( 5 )}{2401}} (- \frac{2401 u + ln ( 5 )}{ln ( 5 )} + 1) = 1

in Lambert form:

\frac{e ^{\frac{2401 u + l n ( 5 )}{2401}} u}{e ^{\frac{l n ( 5 )}{2401}}} = - \frac{ln ( 5 )}{2401 e ^{\frac{l n ( 5 )}{2401}}}

Solve

\frac{e ^{\frac{2401 u + l n ( 5 )}{2401}} u}{e ^{\frac{l n ( 5 )}{2401}}} = - \frac{ln ( 5 )}{2401 e ^{\frac{l n ( 5 )}{2401}}} : u = W_{- 1} (- \frac{ln ( 5 )}{2401 e ^{\frac{l n ( 5 )}{2401}}}), u = W_{0} (- \frac{ln ( 5 )}{2401 e ^{\frac{l n ( 5 )}{2401}}})

u = W_{- 1} (- \frac{ln ( 5 )}{2401 e ^{\frac{l n ( 5 )}{2401}}}), u = W_{0} (- \frac{ln ( 5 )}{2401 e ^{\frac{l n ( 5 )}{2401}}})

Substitute back

u = (- x - \frac{1}{2401}) ln (5),

solve for

x

Solve

(- x - \frac{1}{2401}) ln (5) = W_{- 1} (- \frac{ln ( 5 )}{2401 e ^{\frac{l n ( 5 )}{2401}}}) : x = - \frac{2401 W _{- 1} ( - \frac{l n ( 5 )}{2401 e ^{\frac{l n ( 5 )}{2401}}} ) + ln ( 5 )}{2401 ln ( 5 )}

Solve

(- x - \frac{1}{2401}) ln (5) = W_{0} (- \frac{ln ( 5 )}{2401 e ^{\frac{l n ( 5 )}{2401}}}) : x = - \frac{2401 W _{0} ( - \frac{l n ( 5 )}{2401 e ^{\frac{l n ( 5 )}{2401}}} ) + ln ( 5 )}{2401 ln ( 5 )}

x = - \frac{2401 W _{- 1} ( - \frac{l n ( 5 )}{2401 e ^{\frac{l n ( 5 )}{2401}}} ) + ln ( 5 )}{2401 ln ( 5 )}, x = - \frac{2401 W _{0} ( - \frac{l n ( 5 )}{2401 e ^{\frac{l n ( 5 )}{2401}}} ) + ln ( 5 )}{2401 ln ( 5 )}

Popular Examples

2 = 2^{x}

4^{x} + 1 = 64

2^4*2^{3n-1}=64

2^{4} \cdot 2^{3 n - 1} = 64

2^{a} = 4

2^{a} = 5

Frequently Asked Questions (FAQ)

What is the answer to 7^4x+1=5^x ?
The answer to 7^4x+1=5^x is
Solve for x: 7^4x+1=5^x
The solution is