Question

Lg(x^2-5x+68)=2(lg16-lg2)

Answer 1

Lg16-lg2=lg16/2=lg8  
lg(a/b)=lga-lgb
lg(x^2-5x+68)=lg8^2
2lg8=lg8^2
x^2-5x+68=64
x^2-5x+4=0
x=(5+-√(25-16))/2
x=(5+-3)/2
x1=(5-3)/2=1
x2=(5+3)/2=4

Answer 2

To solve the equation lg(x^2 - 5x + 68) = 2(lg16 - lg2), we can start by simplifying the right side of the equation using the properties of logarithms.

lg16 - lg2 can be simplified as follows: lg16 - lg2 = lg(16/2) = lg8

Now, let's substitute this simplified expression back into the equation:

lg(x^2 - 5x + 68) = 2(lg8)

Next, we can use the property of logarithms that states lg(a) = b is equivalent to 10^b = a. Applying this property to both sides of the equation, we have:

x^2 - 5x + 68 = 10^(2lg8)

Now, let's simplify further:

x^2 - 5x + 68 = 10^(lg64)

Since lg64 = 2, we have:

x^2 - 5x + 68 = 10^2

x^2 - 5x + 68 = 100

Rearranging the equation, we have a quadratic equation:

x^2 - 5x + 68 - 100 = 0

x^2 - 5x - 32 = 0

Now we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = -5, and c = -32. Substituting these values into the quadratic formula:

x = (-(-5) ± √((-5)^2 - 4(1)(-32))) / (2(1))

Simplifying further:

x = (5 ± √(25 + 128)) / 2

x = (5 ± √153) / 2

Therefore, the solutions to the equation lg(x^2 - 5x + 68) = 2(lg16 - lg2) are:

x = (5 + √153) / 2 and x = (5 - √153) / 2