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

The answer to x^{2*log_{10}(x)}=10x^2 is x=10^{(1+sqrt(3))/2},x=10^{(1-sqrt(3))/2}

Solve for x: x^{2*log_{10}(x)}=10x^2

The solution is x=10^{(1+sqrt(3))/2},x=10^{(1-sqrt(3))/2}

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

x = 1 0^{\frac{1 + 3}{2}}, x = 1 0^{\frac{1 - 3}{2}}

Decimal Notation

x = 23.22872 \dots, x = 0.43050 \dots

Solution steps

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

Apply log rules

2 lo g_{10} (x) lo g_{10} (x) = 2 lo g_{10} (x) + 1

Rewrite the equation with

lo g_{10} (x) = u

2 uu = 2 u + 1

Solve

2 uu = 2 u + 1 : u = \frac{1 + 3}{2}, u = \frac{1 - 3}{2}

u = \frac{1 + 3}{2}, u = \frac{1 - 3}{2}

Substitute back

u = lo g_{10} (x),

solve for

x

Solve

lo g_{10} (x) = \frac{1 + 3}{2} : x = 1 0^{\frac{1 + 3}{2}}

Solve

lo g_{10} (x) = \frac{1 - 3}{2} : x = 1 0^{\frac{1 - 3}{2}}

x = 1 0^{\frac{1 + 3}{2}}, x = 1 0^{\frac{1 - 3}{2}}

Verify Solutions:

x = 1 0^{\frac{1 + 3}{2}}

True

, x = 1 0^{\frac{1 - 3}{2}}

True

The solutions are

x = 1 0^{\frac{1 + 3}{2}}, x = 1 0^{\frac{1 - 3}{2}}

\frac{lo g _{2} ( x ) - 1}{2} = lo g_{2} (x)

lo g_{10} (x) + lo g_{10} (x + 1) = 2

lo g_{10} (2 x - 3) = 1

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

lo g_{10} ((5 x - 3)^{2}) + lo g_{10} ((2 x + 3)^{3}) = 2

What is the answer to x^{2*log_{10}(x)}=10x^2 ?
The answer to x^{2*log_{10}(x)}=10x^2 is x=10^{(1+sqrt(3))/2},x=10^{(1-sqrt(3))/2}
Solve for x: x^{2*log_{10}(x)}=10x^2
The solution is x=10^{(1+sqrt(3))/2},x=10^{(1-sqrt(3))/2}