Step-by-step solution of a logarithmic equation leading to the solution set {5}.

Let's carefully analyze the problem and solution provided in your image.

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Problem:

Solve the logarithmic equation:

$$
\log_2 5 + \log_2 x = \log_2(x + 20)
$$

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Step-by-Step Solution:

#### Step 1: Use Logarithmic Properties

We use the product rule of logarithms:

$$
\log_b A + \log_b B = \log_b(AB)
$$

Apply this to the left-hand side:

$$
\log_2 5 + \log_2 x = \log_2(5x)
$$

So the equation becomes:

$$
\log_2(5x) = \log_2(x + 20)
$$

#### Step 2: Eliminate Logarithms

If $\log_b A = \log_b B$, then $A = B$ (since log is one-to-one).

So:

$$
5x = x + 20
$$

#### Step 3: Solve for $x$

$$
5x - x = 20 \\
4x = 20 \\
x = 5
$$

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Step 4: Check the Solution

Plug $x = 5$ back into the original equation:

Left-hand side:
$$
\log_2 5 + \log_2 5 = 2\log_2 5
$$

Right-hand side:
$$
\log_2(5 + 20) = \log_2(25)
$$

Now, note that:
$$
2\log_2 5 = \log_2(5^2) = \log_2(25)
$$

✔ So both sides are equal. The solution checks out.

Also, check the domain:
- $x > 0$ because $\log_2 x$ is only defined for $x > 0$
- $x + 20 > 0$ → always true if $x > 0$

Since $x = 5 > 0$, it's valid.

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✔ Final Answer:

$$
\boxed{x = 5}
$$

The solution set is $\boxed{\{5\}}$

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✔ Summary:

- Used logarithmic identity: $\log_a A + \log_a B = \log_a(AB)$
- Equated arguments after logs were removed
- Solved linear equation
- Verified solution by substitution
- Confirmed domain is satisfied

✔ All steps are correct! The handwritten solution is accurate.