Solving Logarithmic Equations: The Solution to log2 (3x – 7) = 3
Logarithmic equations can be tricky to solve, and often require careful manipulation of the equation in order to arrive at the solution. In this post, we will walk through the steps needed to solve the equation log2 (3x – 7) = 3.
Step 1: Identify the Base
The first step in solving any logarithmic equation is to identify the base of the logarithm. In this case, the base is 2, since the logarithm is written as log2.
Step 2: Rewrite the Equation
Next, we need to rewrite the equation in exponential form. To do this, we raise both sides of the equation to the power of the base, which is 2 in this case. This gives us:
2^log2 (3x – 7) = 2^3
Simplifying the left side of the equation using the rule that 2^log2 x = x, we get:
3x – 7 = 8
Step 3: Solve for x
Now that we have reduced the equation to a simple linear equation, we can solve it for x. Adding 7 to both sides of the equation, we get:
3x = 15
Dividing both sides by 3, we finally arrive at the solution:
x = 5
So there you have it – the solution to the equation log2 (3x – 7) = 3 is x = 5. While solving logarithmic equations can be challenging, with a little practice and the right techniques, anyone can master them.