Solve this log function:
a) log (sqrt(x^2-3x))=0.5
b) log (sqrt(x^2+48x))=1
a) Assuming that you mean log to base 10,
sqrt(x^2-3x) = 10^0.5 = sqrt 10
x^2 -3x = 10
(x-5)(x+2) = 0
x= 5 or -2
Do (b) the same way. First restate it without the log function.
sqrt (x^2 + 48x) = 10
x^2 + 48x = 100
Then solve the quadratic function.
a) Assuming that you mean log to base 10,
sqrt(x^2-3x) = 10^0.5 = sqrt 10
x^2 -3x = 10
(x-5)(x+2) = 0
x= 5 or -2
Do (b) the same way. First restate it without the log function.
sqrt (x^2 + 48x) = 10
Then solve the quadratic that results when you square that.
We seem to have a faulty web server. It took several minutes for my answer to be posted, so I did it again
To solve each of these logarithmic equations, we can use logarithm properties and algebraic techniques. Let's solve each equation step by step:
a) log (sqrt(x^2 - 3x)) = 0.5
Step 1: Rewrite the square root as an exponent:
sqrt(x^2 - 3x) = 10^(0.5)
Step 2: Square both sides of the equation to eliminate the square root:
x^2 - 3x = (10^(0.5))^2
Simplifying the right side:
x^2 - 3x = 10
Step 3: Move all terms to one side of the equation to set it equal to zero:
x^2 - 3x - 10 = 0
Step 4: Factor the quadratic equation:
(x - 5)(x + 2) = 0
Step 5: Set each factor equal to zero and solve for x:
x - 5 = 0 --> x = 5
x + 2 = 0 --> x = -2
Therefore, the solutions to the equation are x = 5 and x = -2.
b) log (sqrt(x^2 + 48x)) = 1
Step 1: Rewrite the square root as an exponent:
sqrt(x^2 + 48x) = 10^1
Step 2: Square both sides of the equation to eliminate the square root:
x^2 + 48x = 10^2
Simplifying the right side:
x^2 + 48x = 100
Step 3: Move all terms to one side of the equation to set it equal to zero:
x^2 + 48x - 100 = 0
Step 4: Use factoring or the quadratic formula to solve for x:
This quadratic equation cannot be easily factored, so we will use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
For our equation:
a = 1
b = 48
c = -100
Using the quadratic formula, we can calculate the solutions for x.
x = (-48 ± sqrt(48^2 - 4(1)(-100))) / (2(1))
After simplifying the equation, we get:
x = (-48 ± sqrt(2304 + 400)) / 2
x = (-48 ± sqrt(2704)) / 2
x = (-48 ± 52) / 2
This leads to two possible solutions:
x = (-48 + 52) / 2 --> x = 2/2 --> x = 1
x = (-48 - 52) / 2 --> x = -100/2 -> x = -50
Therefore, the solutions to the equation are x = 1 and x = -50.