how do i do:
log(base10)(c^2-1)-2 = log(base10)(c+1)
To solve the equation log(base10)(c^2-1) - 2 = log(base10)(c+1), we can simplify it step by step. Here's how you can approach it:
Step 1: Combine the logarithms on both sides of the equation using the properties of logarithms. The property we will use is:
log(basea)(b) - log(basea)(c) = log(basea)(b/c)
Applying this property to our equation, we have:
log(base10)((c^2-1)/(10^2)) = log(base10)(c+1)
Step 2: Simplify the equation further by eliminating the logarithm on both sides. Since the bases of both logarithms are the same (base 10 in this case), we can eliminate the logarithms and equate the arguments:
(c^2-1)/(10^2) = c+1
Now, we have a quadratic equation which we will solve in the next step.
Step 3: Simplify the left side of the equation:
(c^2-1)/100 = c+1
Step 4: Multiply both sides of the equation by 100 to eliminate the fraction:
c^2 - 1 = 100(c + 1)
Step 5: Expand the equation:
c^2 - 1 = 100c + 100
Step 6: Rearrange the equation to the standard quadratic form by bringing all terms to one side:
c^2 - 100c - 101 = 0
Step 7: Now, we can solve the quadratic equation. There are multiple ways to solve it, but let's use the quadratic formula:
c = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -100, and c = -101.
Plugging these values into the quadratic formula, we have:
c = (-(-100) ± √((-100)^2 - 4(1)(-101))) / (2(1))
Simplifying further:
c = (100 ± √(10000 + 404))/2
c = (100 ± √10404)/2
c = (100 ± 102)/2
Step 8: Solve for c:
c = (100 + 102)/2 or c = (100 - 102)/2
Simplifying:
c = 202/2 or c = -2/2
c = 101 or c = -1
So, the solutions to the equation log(base10)(c^2-1) - 2 = log(base10)(c+1) are c = 101 and c = -1.