The expression you provided appears to be a logarithmic one:

\[
2 \cdot \log_2(x) + \log_2(x + 5)
\]

Here’s how you can simplify it step-by-step:

### Step 1: Use the logarithmic property for multiplication.
The property of logarithms we’ll use is:
\[
a \cdot \log_b(c) = \log_b(c^a)
\]
So, the first term \( 2 \cdot \log_2(x) \) becomes:
\[
\log_2(x^2)
\]
Thus, the expression becomes:
\[
\log_2(x^2) + \log_2(x + 5)
\]

### Step 2: Use the logarithmic property for addition.
The property of logarithms we’ll use now is:
\[
\log_b(a) + \log_b(c) = \log_b(a \cdot c)
\]
Applying this to our expression:
\[
\log_2(x^2) + \log_2(x + 5) = \log_2(x^2 \cdot (x + 5))
\]

### Step 3: Simplify the product.
Now, multiply \( x^2 \) by \( x + 5 \):
\[
x^2 \cdot (x + 5) = x^3 + 5x^2
\]
Thus, the expression simplifies to:
\[
\log_2(x^3 + 5x^2)
\]

### Final Answer:
\[
\log_2(x^3 + 5x^2)
\]

Would you like a more detailed explanation or have any questions?

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Here are 5 related questions:
1. What are the properties of logarithms used in this simplification?
2. How can we solve logarithmic equations like the one given?
3. What is the difference between natural logarithms and base-2 logarithms?
4. How do you graph logarithmic functions?
5. How would you solve \( \log_2(x^3 + 5x^2) = y \) for \( x \)?

**Tip:** Always remember that when adding or multiplying logarithms, the base of the logarithm must be the same for these rules to apply.

Learn how to simplify the logarithmic expression 2 * log_2(x) + log_2(x + 5) using properties of logarithms. This step-by-step solution covers key algebraic concepts suitable for students in Grades 9-12.

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