solve the following equation show all work log4(x-4)= -1.
by definition:
log4 (x-4) = -1
is equivalent to
x-4 = 4^-1
x-4 = 1/4
x = 1/4 - 4
= -15/4
so we would have log (-3/4-4)
but we cannot take the log of a negative, so ....
no solution.
log takes on all real values for its range. So, something's wrong here.
x - 4 = 1/4
x = 1/4 + 4
x = 17/4
log4(17/4 - 4) = log4(1/4) = -1
To solve the equation log4(x - 4) = -1, you need to use the basic properties of logarithms.
Step 1: Rewrite the equation using exponential form. In exponential form, loga(b) = c is equivalent to a^c = b. Applying this to our equation:
4^(-1) = x - 4
Step 2: Simplify the equation by evaluating the exponential expression on the left side:
1/4 = x - 4
Step 3: Isolate the variable x by adding 4 to both sides of the equation:
1/4 + 4 = x
Step 4: Simplify the expression on the left side by finding a common denominator:
1/4 + 16/4 = x
17/4 = x
The solution to the equation log4(x - 4) = -1 is x = 17/4.
Note: It's important to verify if the solution satisfies the original equation since we took the logarithm of both sides. Therefore, substitute the value x = 17/4 back into the original equation to ensure it holds true.