Could someone please help me on this problem. I'm really confused, and this assignment is due tomorrow.
Solve the logarithmic equation algebraically. Approximate the result to three decimal places.
log4x-log4(x-1)=1/2
To start off the equation, I know you would do the following:
log4(x/x-1)=1/2
After that, I'm just not sure how to solve it. Any help would be greatly appreciated.
log4 ( ) = 1/2
or ( )=4^1/2
or ( )=2
2x-2=x
x=2
To solve the given logarithmic equation algebraically, we can follow these steps:
Step 1: Combine the logarithmic expressions on the left side of the equation using the properties of logarithms. In this case, since the bases are the same (both are 4), we can use the quotient rule of logarithms, which states that log base b of x minus log base b of y is equal to log base b of (x/y):
log4(x/(x-1)) = 1/2
Step 2: Convert the equation to exponential form. The logarithmic equation log base b of x = y is equivalent to b^y = x. In this case, since the base is 4, we have:
4^(1/2) = x/(x-1)
Step 3: Simplify the equation by squaring both sides:
(4^(1/2))^2 = (x/(x-1))^2
4 = (x/(x-1))^2
Step 4: Solve for x by taking the square root of both sides:
2 = x/(x-1)
Step 5: Multiply both sides of the equation by (x-1) to eliminate the fraction:
2(x-1) = x
Step 6: Distribute and simplify:
2x - 2 = x
Step 7: Subtract x from both sides:
x - 2 = 0
Step 8: Add 2 to both sides:
x = 2
So, the solution to the logarithmic equation log4x - log4(x-1) = 1/2 is x = 2.
To approximate the result to three decimal places, we round the answer to three digits after the decimal point: x ≈ 2.000.