Answer:
x =2
Step-by-step explanation:
Solve for x:
log(x + 3) = log(10) - log(x)
Hint: | Move everything to the left hand side.
Subtract log(10) - log(x) from both sides:
-log(10) + log(x) + log(x + 3) = 0
Hint: | Combine logarithms.
-log(10) + log(x) + log(x + 3) = log(1/10) + log(x) + log(x + 3) = log(1/10 x (x + 3)):
log(1/10 x (x + 3)) = 0
Hint: | Eliminate the logarithm from the left hand side.
Cancel logarithms by taking exp of both sides:
1/10 x (x + 3) = 1
Hint: | Multiply both sides by a constant to simplify the equation.
Multiply both sides by 10:
x (x + 3) = 10
Hint: | Write the quadratic polynomial on the left hand side in standard form.
Expand out terms of the left hand side:
x^2 + 3 x = 10
Hint: | Take one half of the coefficient of x and square it, then add it to both sides.
Add 9/4 to both sides:
x^2 + 3 x + 9/4 = 49/4
Hint: | Factor the left hand side.
Write the left hand side as a square:
(x + 3/2)^2 = 49/4
Hint: | Eliminate the exponent on the left hand side.
Take the square root of both sides:
x + 3/2 = 7/2 or x + 3/2 = -7/2
Hint: | Look at the first equation: Solve for x.
Subtract 3/2 from both sides:
x = 2 or x + 3/2 = -7/2
Hint: | Look at the second equation: Solve for x.
Subtract 3/2 from both sides:
x = 2 or x = -5
Hint: | Now test that these solutions are correct by substituting into the original equation.
Check the solution x = -5.
log(x + 3) ⇒ log(3 - 5) = i π + log(2) ≈ 0.693147 + 3.14159 i
log(10) - log(x) ⇒ log(10) - log(-5) = -i π + log(2) ≈ 0.693147 - 3.14159 i:
So this solution is incorrect
Hint: | Check the solution x = 2.
log(x + 3) ⇒ log(3 + 2) = log(5) ≈ 1.60944
log(10) - log(x) ⇒ log(10) - log(2) = log(5) ≈ 1.60944:
So this solution is correct
Hint: | Gather any correct solutions.
The solution is:
Answer: x = 2
