To solve the equation $\log_2(x) - 7 = \log_2(3)$, follow these steps:

Step 1: Isolate $\log_2(x)$

Add $7$ to both sides of the equation: $\log_2(x) = \log_2(3) + 7$

Step 2: Rewrite $7$ in base $2$ logarithmic form

Using the property $a = \log_b(b^a)$: $7 = \log_2(2^7)$ Thus, the equation becomes: $\log_2(x) = \log_2(3) + \log_2(2^7)$

Step 3: Apply the logarithmic addition property

Using $\log_b(A) + \log_b(B) = \log_b(A \cdot B)$, combine the terms: $\log_2(x) = \log_2(3 \cdot 2^7)$

Step 4: Simplify the argument

$3 \cdot 2^7 = 3 \cdot 128 = 384$ So: $\log_2(x) = \log_2(384)$

Step 5: Solve for $x$

Since $\log_2(x) = \log_2(384)$, the solution is: $x = 384$

Final Answer:

$\boxed{x = 384}$

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Would you like further details or have questions?

Related Questions:

1. What are the logarithmic properties used in this solution?
2. How do you convert between different logarithmic bases?
3. Can this problem be solved without logarithmic properties?
4. What is the general form of solving $\log_b(x) = c$?
5. How is the value $2^7$ calculated efficiently?

Tip:

When solving logarithmic equations, remember to check the domain of the logarithm to ensure all values are valid (e.g., $x > 0$ in $\log(x)$).