The equation in the image is:

$\log_2(8) - \log_2(\sqrt{2x}) = 0.5$

Let’s solve this step by step:

Step 1: Simplify the logarithmic terms

1. The logarithm of $8$ with base 2 can be simplified: \log_2(8) = 3 \quad \text{(since 2^3 = 8)}.

2. Rewrite the square root term using the property of logarithms: $\log_2(\sqrt{2x}) = \log_2((2x)^{1/2}).$ Using the logarithmic exponent rule $\log_b(a^c) = c\log_b(a)$: $\log_2(\sqrt{2x}) = \frac{1}{2} \log_2(2x).$

Thus, the equation becomes: $3 - \frac{1}{2} \log_2(2x) = 0.5.$

Step 2: Isolate the logarithmic term

Move $3$ to the right side: $-\frac{1}{2} \log_2(2x) = 0.5 - 3.$ Simplify the right-hand side: $-\frac{1}{2} \log_2(2x) = -2.5.$

Multiply through by $-2$ to eliminate the fraction: $\log_2(2x) = 5.$

Step 3: Eliminate the logarithm

Convert the logarithmic equation to its exponential form. Recall that $\log_b(a) = c \implies b^c = a$. Here: $\log_2(2x) = 5 \implies 2^5 = 2x.$ Simplify $2^5$: $32 = 2x.$

Step 4: Solve for $x$

Divide both sides by 2: $x = \frac{32}{2}.$ $x = 16.$

Final Answer:

The solution is: $\boxed{x = 16}.$

Let me know if you need further clarification or additional details!

Related Questions:

1. What are the logarithmic properties used to simplify terms?
2. How do you convert logarithmic form to exponential form?
3. How do you solve $\log_b(x) = c$ for $x$?
4. What is the general rule for $\log_b(a^c)$?
5. How do you solve logarithmic equations involving square roots?

Tip:

Always simplify logarithmic expressions step by step using their properties, such as the exponent, product, and quotient rules.