Certainly! Let's solve the given equation step-by-step.
The equation is:
[tex]\[ \log_2 (2z + 1) = 2 + \log_2 2 \][/tex]
### Step 1: Simplify the right-hand side
First, we simplify the term on the right-hand side of the equation.
We know that:
[tex]\[ \log_2 2 = 1 \][/tex]
So the equation simplifies to:
[tex]\[ \log_2 (2z + 1) = 2 + 1 \][/tex]
[tex]\[ \log_2 (2z + 1) = 3 \][/tex]
### Step 2: Convert the logarithmic equation to an exponential equation
To remove the logarithm, we convert the logarithmic equation to its exponential form:
[tex]\[ 2z + 1 = 2^3 \][/tex]
### Step 3: Solve the exponential equation
Next, we simplify the right-hand side:
[tex]\[ 2z + 1 = 8 \][/tex]
Subtract 1 from both sides:
[tex]\[ 2z = 7 \][/tex]
Divide both sides by 2:
[tex]\[ z = \frac{7}{2} \][/tex]
### Conclusion
The solution to the equation [tex]\(\log_2 (2z + 1) = 2 + \log_2 2\)[/tex] is:
[tex]\[ z = \frac{7}{2} \][/tex]
So, the value of [tex]\(z\)[/tex] is:
[tex]\[ z = 3.5 \][/tex]
