What is the answer to log_{3}(x^{log_{3}(x)+1})=2 ?

The answer to log_{3}(x^{log_{3}(x)+1})=2 is x=3,x= 1/9

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log_{3}(x^{log_{3}(x)+1})=2

lo g_{3} (x^{l o g_{3} (x) + 1}) = 2

x = 3, x = \frac{1}{9}

Decimal Notation

x = 3, x = 0.11111 \dots

Solution steps

lo g_{3} (x^{l o g_{3} (x) + 1}) = 2

Apply exponent rules

lo g_{3} (e^{(l o g_{3} (x) + 1) l n (x)}) = 2

Rewrite the equation with

e^{(l o g_{3} (x) + 1) l n (x)} = u

lo g_{3} (u) = 2

Solve

lo g_{3} (u) = 2 : u = 9

u = 9

Substitute back

u = e^{(l o g_{3} (x) + 1) l n (x)},

solve for

x

Solve

e^{(l o g_{3} (x) + 1) l n (x)} = 9 : x = 3, x = \frac{1}{9}

x = 3, x = \frac{1}{9}

Verify Solutions:

x = 3

True

, x = \frac{1}{9}

True

The solutions are

x = 3, x = \frac{1}{9}

5320 e^{x} = 14756

e^{x} + 6 e^{- x} - 5 = 0

57 = 1 5^{x}

x, x y - e^{x} + e^{y} = 0

x, 5 (2 5^{x}) = 3.12 5^{y}

What is the answer to log_{3}(x^{log_{3}(x)+1})=2 ?
The answer to log_{3}(x^{log_{3}(x)+1})=2 is x=3,x= 1/9
Solve for x: log_{3}(x^{log_{3}(x)+1})=2
The solution is x=3,x= 1/9