Solve the following logarithm equation. Please show all of your work.
log (5 + x) – log(x – 2) = log 2
log (5 + x) – log(x – 2) = log 2
log [(5+x)/(x-2)] = log 2
(5+x)/(x-2) = 2
2x - 4 = 5+x
x = 9
Use law of logarithms,
log(a)-log(b)=log(a/b)
So
log (5 + x) – log(x – 2) = log 2
=log((5+x)/(x-2))=log2
Take anti-log on both sides
(5-x)/(x-2)=2
(5-x)=2(x-2)
Solve for x.
(5+x)/(x-2)=2
5+x=2*(x-2)
5+x=2x-4
5+4=2x-x
9=x
x=9
−6ln(x) − 8ln(y) +
1
8
ln(z)
To solve the logarithm equation: log(5 + x) - log(x - 2) = log 2, we can use the properties of logarithms to simplify the equation.
Step 1: Combine the logarithms on the left side using the quotient rule.
log((5 + x)/(x - 2)) = log 2
Step 2: Remove the logarithms by converting the equation to exponential form.
(5 + x)/(x - 2) = 2
Step 3: Multiply both sides of the equation by (x - 2) to eliminate the fraction.
(5 + x) = 2(x - 2)
Step 4: Distribute the 2 on the right side.
5 + x = 2x - 4
Step 5: Move all the x terms to one side and the constant terms to the other side.
x - 2x = -4 - 5
-x = -9
Step 6: Multiply both sides of the equation by -1 to isolate x.
x = 9
Step 7: Check the solution by substituting x back into the original equation.
log(5 + x) - log(x - 2) = log 2
log(5 + 9) - log(9 - 2) = log 2
log(14) - log(7) = log 2
log(14/7) = log 2
log(2) = log 2 (True)
Therefore, the solution to the logarithm equation log(5 + x) - log(x - 2) = log 2 is x = 9.