To solve the equation $\log_2(4x + 6) = 2$, we will proceed as follows:

Step 1: Rewrite the logarithmic equation in exponential form

The general property of logarithms is: $\log_b(a) = c \implies b^c = a$ Here: $\log_2(4x + 6) = 2 \implies 2^2 = 4x + 6$

Step 2: Simplify the exponential expression

$2^2 = 4 \implies 4 = 4x + 6$

Step 3: Solve for $x$

$4 = 4x + 6$ Subtract 6 from both sides: $4 - 6 = 4x$ $-2 = 4x$ Divide both sides by 4: $x = -\frac{1}{2}$

Final Answer:

$x = -\frac{1}{2}$

Would you like additional explanations or details on logarithmic properties? Here are related questions for further exploration:

1. What are the properties of logarithms, such as product, quotient, and power rules?
2. How do you verify solutions of logarithmic equations?
3. What is the difference between natural logarithms ($\ln$) and logarithms with other bases?
4. Can a logarithmic equation have no solution? Why or why not?
5. How does the base of a logarithm affect the growth of the function?

Tip: Always check the domain of the logarithmic function. For $\log_b(x)$, $x$ must be positive ($x > 0$).