What is the answer to x^{log_{10}(x)}=1000x^2 ?

The answer to x^{log_{10}(x)}=1000x^2 is x=1000,x= 1/10

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x^{log_{10}(x)}=1000x^2

x^{l o g_{10} (x)} = 1000 x^{2}

x = 1000, x = \frac{1}{10}

Solution steps

x^{l o g_{10} (x)} = 1000 x^{2}

Apply log rules

lo g_{10} (x) lo g_{10} (x) = 2 lo g_{10} (x) + 3

Rewrite the equation with

lo g_{10} (x) = u

uu = 2 u + 3

Solve

uu = 2 u + 3 : u = 3, u = - 1

u = 3, u = - 1

Substitute back

u = lo g_{10} (x),

solve for

x

Solve

lo g_{10} (x) = 3 : x = 1000

Solve

lo g_{10} (x) = - 1 : x = \frac{1}{10}

x = 1000, x = \frac{1}{10}

Verify Solutions:

x = 1000

True

, x = \frac{1}{10}

True

The solutions are

x = 1000, x = \frac{1}{10}

6 - 5 ln (x) = 7

- 2 + lo g_{3} (x - 8) = lo g_{3} (3)

lo g_{2} (k^{3}) = 6

lo g_{10} (2 x) = 6

lo g_{10} (2 x) = 5

What is the answer to x^{log_{10}(x)}=1000x^2 ?
The answer to x^{log_{10}(x)}=1000x^2 is x=1000,x= 1/10
Solve for x: x^{log_{10}(x)}=1000x^2
The solution is x=1000,x= 1/10