What are the solutions to the equation log_{10}(4)(x+2)-log_{10}(16)(x^2-4)=1 ?

The solutions to the equation log_{10}(4)(x+2)-log_{10}(16)(x^2-4)=1 are

Find the zeros of log_{10}(4)(x+2)-log_{10}(16)(x^2-4)=1

The zeros of log_{10}(4)(x+2)-log_{10}(16)(x^2-4)=1 are

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lo g_{10} (4) (x + 2) - lo g_{10} (16) (x^{2} - 4) = 1

x = - \frac{- 2 lo g _{10} ( 2 ) + 324 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{8 lo g _{10} ( 2 )}, x = \frac{2 lo g _{10} ( 2 ) + 324 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{8 lo g _{10} ( 2 )}

Decimal Notation

x = - 1.80718 \dots, x = 2.30718 \dots

Solution steps

lo g_{10} (4) (x + 2) - lo g_{10} (16) (x^{2} - 4) = 1

Expand

lo g_{10} (4) (x + 2) - lo g_{10} (16) (x^{2} - 4) : - 4 lo g_{10} (2) x^{2} + 2 lo g_{10} (2) x + 20 lo g_{10} (2)

- 4 lo g_{10} (2) x^{2} + 2 lo g_{10} (2) x + 20 lo g_{10} (2) = 1

Move

1

to the left side

- 4 lo g_{10} (2) x^{2} + 2 lo g_{10} (2) x + 20 lo g_{10} (2) - 1 = 0

Solve with the quadratic formula

x_{1, 2} = \frac{- 2 lo g _{10} ( 2 ) \pm ( 2 lo g _{10} ( 2 ) ) ^{2} - 4 ( - 4 lo g _{10} ( 2 ) ) ( 20 lo g _{10} ( 2 ) - 1 )}{2 ( - 4 lo g _{10} ( 2 ) )}

(2 lo g_{10} (2))^{2} - 4 (- 4 lo g_{10} (2)) (20 lo g_{10} (2) - 1) = 324 lo g_{10} (2)^{2} - 16 lo g_{10} (2)

x_{1, 2} = \frac{- 2 lo g _{10} ( 2 ) \pm 324 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{2 ( - 4 lo g _{10} ( 2 ) )}

Separate the solutions

x_{1} = \frac{- 2 lo g _{10} ( 2 ) + 324 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{2 ( - 4 lo g _{10} ( 2 ) )}, x_{2} = \frac{- 2 lo g _{10} ( 2 ) - 324 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{2 ( - 4 lo g _{10} ( 2 ) )}

x = \frac{- 2 lo g _{10} ( 2 ) + 324 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{2 ( - 4 lo g _{10} ( 2 ) )} : - \frac{- 2 lo g _{10} ( 2 ) + 324 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{8 lo g _{10} ( 2 )}

x = \frac{- 2 lo g _{10} ( 2 ) - 324 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{2 ( - 4 lo g _{10} ( 2 ) )} : \frac{2 lo g _{10} ( 2 ) + 324 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{8 lo g _{10} ( 2 )}

The solutions to the quadratic equation are:

x = - \frac{- 2 lo g _{10} ( 2 ) + 324 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{8 lo g _{10} ( 2 )}, x = \frac{2 lo g _{10} ( 2 ) + 324 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{8 lo g _{10} ( 2 )}

2^{2} + b^{2} = 6^{2}

x^{2} + 6 x - 280 = 0

x, x^{2} - 4 y^{2} + 9 z^{2} = 36

x^{2} + 2 a x - 35 a^{2} = 0

k^{2} - 4 k - 21 = 0

What are the solutions to the equation log_{10}(4)(x+2)-log_{10}(16)(x^2-4)=1 ?
The solutions to the equation log_{10}(4)(x+2)-log_{10}(16)(x^2-4)=1 are
Find the zeros of log_{10}(4)(x+2)-log_{10}(16)(x^2-4)=1
The zeros of log_{10}(4)(x+2)-log_{10}(16)(x^2-4)=1 are