What is the answer to 1.2x=1.2^x ?

The answer to 1.2x=1.2^x is x=-(\W_{-1}(-\frac{ln(1.2))/(1.2))}{ln(1.2)},x=1

Solve for x: 1.2x=1.2^x

The solution is x=-(\W_{-1}(-\frac{ln(1.2))/(1.2))}{ln(1.2)},x=1

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1.2 x = 1. 2^{x}

x = - \frac{W _{- 1} ( - \frac{l n ( 1.2 )}{1.2} )}{ln ( 1.2 )}, x = 1

Decimal Notation

x = 16.31359 \dots, x = 1

Solution steps

1.2 x = 1. 2^{x}

Prepare

1.2 x = 1. 2^{x}

for Lambert form:

e^{l n (1.2) (1 - x)} x = 1

Rewrite the equation with

- x ln (1.2) = u

and

x = - \frac{u}{ln ( 1.2 )}

e^{l n (1.2) (1 - (- \frac{u}{l n ( 1.2 )}))} (- \frac{u}{ln ( 1.2 )}) = 1

Simplify

- \frac{e ^{l n (1.2) + u} u}{ln ( 1.2 )} = 1

Rewrite

- \frac{e ^{l n (1.2) + u} u}{ln ( 1.2 )} = 1

in Lambert form:

e^{u} u = - \frac{ln ( 1.2 )}{1.2}

Solve

e^{u} u = - \frac{ln ( 1.2 )}{1.2} : u = W_{- 1} (- \frac{ln ( 1.2 )}{1.2}), u = - ln (1.2)

u = W_{- 1} (- \frac{ln ( 1.2 )}{1.2}), u = - ln (1.2)

Substitute back

u = - x ln (1.2),

solve for

x

Solve

- x ln (1.2) = W_{- 1} (- \frac{ln ( 1.2 )}{1.2}) : x = - \frac{W _{- 1} ( - \frac{l n ( 1.2 )}{1.2} )}{ln ( 1.2 )}

Solve

- x ln (1.2) = - ln (1.2) : x = 1

x = - \frac{W _{- 1} ( - \frac{l n ( 1.2 )}{1.2} )}{ln ( 1.2 )}, x = 1

1 6^{x} + 3 \cdot 4^{x} - 10 = 0

2^{x} - 8 = 0

3^{2 x} - 5 (3^{x}) = - 6

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

7 e^{13 x} = 20

What is the answer to 1.2x=1.2^x ?
The answer to 1.2x=1.2^x is x=-(\W_{-1}(-\frac{ln(1.2))/(1.2))}{ln(1.2)},x=1
Solve for x: 1.2x=1.2^x
The solution is x=-(\W_{-1}(-\frac{ln(1.2))/(1.2))}{ln(1.2)},x=1