What are the solutions to the equation log_{10}(8)+(x^2-5x+7)log_{10}(3)=log_{10}(24) ?

The solutions to the equation log_{10}(8)+(x^2-5x+7)log_{10}(3)=log_{10}(24) are x=3,x=2

Find the zeros of log_{10}(8)+(x^2-5x+7)log_{10}(3)=log_{10}(24)

The zeros of log_{10}(8)+(x^2-5x+7)log_{10}(3)=log_{10}(24) are x=3,x=2

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lo g_{10} (8) + (x^{2} - 5 x + 7) lo g_{10} (3) = lo g_{10} (24)

x = 3, x = 2

Solution steps

lo g_{10} (8) + (x^{2} - 5 x + 7) lo g_{10} (3) = lo g_{10} (24)

Expand

lo g_{10} (8) + (x^{2} - 5 x + 7) lo g_{10} (3) : 3 lo g_{10} (2) + lo g_{10} (3) x^{2} - 5 lo g_{10} (3) x + 7 lo g_{10} (3)

3 lo g_{10} (2) + lo g_{10} (3) x^{2} - 5 lo g_{10} (3) x + 7 lo g_{10} (3) = lo g_{10} (24)

Move

lo g_{10} (24)

to the left side

6 lo g_{10} (3) + lo g_{10} (3) x^{2} - 5 lo g_{10} (3) x = 0

Write in the standard form

a x^{2} + b x + c = 0

lo g_{10} (3) x^{2} - 5 lo g_{10} (3) x + 6 lo g_{10} (3) = 0

Solve with the quadratic formula

x_{1, 2} = \frac{- ( - 5 lo g _{10} ( 3 ) ) \pm ( - 5 lo g _{10} ( 3 ) ) ^{2} - 4 lo g _{10} ( 3 ) \cdot 6 lo g _{10} ( 3 )}{2 lo g _{10} ( 3 )}

(- 5 lo g_{10} (3))^{2} - 4 lo g_{10} (3) \cdot 6 lo g_{10} (3) = lo g_{10} (3)

x_{1, 2} = \frac{- ( - 5 lo g _{10} ( 3 ) ) \pm lo g _{10} ( 3 )}{2 lo g _{10} ( 3 )}

Separate the solutions

x_{1} = \frac{- ( - 5 lo g _{10} ( 3 ) ) + lo g _{10} ( 3 )}{2 lo g _{10} ( 3 )}, x_{2} = \frac{- ( - 5 lo g _{10} ( 3 ) ) - lo g _{10} ( 3 )}{2 lo g _{10} ( 3 )}

x = \frac{- ( - 5 lo g _{10} ( 3 ) ) + lo g _{10} ( 3 )}{2 lo g _{10} ( 3 )} : 3

x = \frac{- ( - 5 lo g _{10} ( 3 ) ) - lo g _{10} ( 3 )}{2 lo g _{10} ( 3 )} : 2

The solutions to the quadratic equation are:

x = 3, x = 2

3 x^{2} + 24 x + 21 = 0

(x - 7) (x - 4) = 0

5 x^{2} + 3 x = 2

(x - 9)^{2} = 49

(2 x - 4)^{2} = 0

What are the solutions to the equation log_{10}(8)+(x^2-5x+7)log_{10}(3)=log_{10}(24) ?
The solutions to the equation log_{10}(8)+(x^2-5x+7)log_{10}(3)=log_{10}(24) are x=3,x=2
Find the zeros of log_{10}(8)+(x^2-5x+7)log_{10}(3)=log_{10}(24)
The zeros of log_{10}(8)+(x^2-5x+7)log_{10}(3)=log_{10}(24) are x=3,x=2