What is the answer to exln(x)=(ln(x))/(ex) ?

The answer to exln(x)=(ln(x))/(ex) is x=1,x= 1/e

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exln(x)=(ln(x))/(ex)

e x ln (x) = \frac{ln ( x )}{e x}

x = 1, x = \frac{1}{e}

Decimal Notation

x = 1, x = 0.36787 \dots

Solution steps

e x ln (x) = \frac{ln ( x )}{e x}

Multiply both sides by

e x

e x ln (x) e x = \frac{ln ( x )}{e x} e x

Simplify

e x ln (x) e x : e^{2} x^{2} ln (x)

e^{2} x^{2} ln (x) = ln (x)

Move

ln (x)

to the left side

e^{2} x^{2} ln (x) - ln (x) = 0

Factor

e^{2} x^{2} ln (x) - ln (x) : ln (x) (e^{2} x^{2} - 1)

ln (x) (e^{2} x^{2} - 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

ln (x) = 0 or e^{2} x^{2} - 1 = 0

Solve

ln (x) = 0 : x = 1

Solve

e^{2} x^{2} - 1 = 0 : x = \frac{1}{e}, x = - \frac{1}{e}

Verify Solutions:

x = 1

True

, x = \frac{1}{e}

True

, x = - \frac{1}{e}

False

The solutions are

x = 1, x = \frac{1}{e}

x, ln (y) + ln (x) = c

lo g_{5} (6 x) + 5 = 3

lo g_{4} (x + 3) - lo g_{4} (x - 5) = 2

4 lo g_{10} (2 n) - 7 = - 11

lo g_{10} (x) = 5.6

What is the answer to exln(x)=(ln(x))/(ex) ?
The answer to exln(x)=(ln(x))/(ex) is x=1,x= 1/e