solve the logarithm
log_2 (x -1)- log_2(5x +1) = -3
I got -3, is this correct???????
assuming base 2 for ease of reading,
log(x-1) - log(5x+1) = -3
log[(x-1)/(5x+1)] = -3
(x-1)/(5x+1) = 2^-3 = 1/8
8(x-1) = 5x+1
8x - 8 = 5x + 1
3x = 9
x = 3
log 2 - log 16 = 1 - 4 = -3
To solve the logarithmic equation, you need to eliminate the logarithms and solve for x. Here's the step-by-step explanation:
1. Use the logarithmic property that states: log(a) - log(b) = log(a/b)
So, applying this property to the equation, we have:
log_2(x - 1) - log_2(5x + 1) = log_2((x - 1)/(5x + 1))
2. Use another logarithmic property: If log_a(b) = c, then a^c = b.
In this case, we have:
log_2((x - 1)/(5x + 1)) = -3
This can be rewritten as:
2^(-3) = (x - 1)/(5x + 1)
3. Simplify the left side of the equation:
1/8 = (x - 1)/(5x + 1)
4. Cross multiply:
8(x - 1) = 1(5x + 1)
Simplifying both sides, we get:
8x - 8 = 5x + 1
5. Solve for x by isolating the variables:
8x - 5x = 1 + 8
3x = 9
x = 3
So the solution to the logarithmic equation is x = 3.
To verify if -3 is the correct answer, substitute x = -3 back into the original equation and see if both sides of the equation are equal. If they are equal, then -3 is the correct solution.