State the univers and solve each of the following equation
log(3x-1)-log(3x+1)=log16
To solve the equation log(3x-1)-log(3x+1)=log16, we can combine the logarithmic terms using the quotient rule of logarithms, which states that log(a)-log(b) = log(a/b). Applying this rule, we get:
log((3x-1)/(3x+1)) = log16
Now we can eliminate the logarithm by exponentiating both sides with base 10, which gives:
(3x-1)/(3x+1) = 16
Next, we can cross-multiply and simplify as follows:
3x-1 = 16(3x+1)
3x-1 = 48x+16
-45x = 17
x = -17/45
Therefore, the solution to the equation log(3x-1)-log(3x+1)=log16 is x = -17/45. Note that this solution is valid only if 3x-1 and 3x+1 are both positive (since logarithms are defined only for positive arguments). Checking these conditions, we see that -1/3 < x < 1, so the solution is valid.
To solve the equation log(3x-1) - log(3x+1) = log(16), we will apply the logarithmic properties.
Step 1: Combine the logarithms using the quotient rule:
log((3x-1)/(3x+1)) = log(16)
Step 2: Since the logarithms are equal, the expressions inside the logarithms must be equal as well. So we have:
(3x-1)/(3x+1) = 16
Step 3: Cross multiply and simplify the equation:
3x - 1 = 16(3x + 1)
3x - 1 = 48x + 16
Step 4: Rearrange the equation to gather the x terms on one side and the constants on the other side:
3x - 48x = 16 + 1
-45x = 17
Step 5: Divide both sides of the equation by -45 to solve for x:
x = 17 / -45
Step 6: Simplify the fraction (if possible):
x = -17 / 45
Therefore, the solution to the equation log(3x-1) - log(3x+1) = log(16) is x = -17/45.