Final answer:
To solve the logarithmic equation log(4)(164) = 2x − 6, we convert it to exponential form and solve for x, resulting in the solution x = 4.
Step-by-step explanation:
To solve the equation log(4)(164) = 2x − 6, we first need to understand that the logarithmic equation can be converted into an exponential form. This step is important because it allows us to isolate the variable x. The base of the logarithm is 4, so taking the inverse log or calculating 4 to the power of the right side will give us the number that the log is referring to. Let's solve for x:
Write the logarithmic equation in exponential form: 4^(2x − 6) = 164.
Find the value that makes the equation true.
Simplify to find x: 2x − 6 = log(4)(164).
Use the properties of logarithms to solve for x.
By following these steps, we get:
2x − 6 = log(4)(164)
2x = log(4)(164) + 6
x = (log(4)(164) + 6) / 2
x = (log(4)(164) / 2) + 3
x = (2 / 2) + 3
x = 1 + 3
x = 4
So the solution to the equation is x = 4.