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We are given the equation:

\[
\log_2(5x) - \log_4(2x) = 3
\]

### Step 1: Express \(\log_4(2x)\) in terms of base 2

We can use the change of base formula:

\[
\log_b(a) = \frac{\log_c(a)}{\log_c(b)}
\]

In this case, converting \(\log_4(2x)\) to base 2:

\[
\log_4(2x) = \frac{\log_2(2x)}{\log_2(4)} = \frac{\log_2(2x)}{2}
\]

Thus, the equation becomes:

\[
\log_2(5x) - \frac{1}{2} \log_2(2x) = 3
\]

### Step 2: Simplify the logarithms

Next, apply the logarithmic identity \(\log_b(ab) = \log_b(a) + \log_b(b)\):

\[
\log_2(5x) = \log_2(5) + \log_2(x)
\]
\[
\log_2(2x) = \log_2(2) + \log_2(x) = 1 + \log_2(x)
\]

Substituting these into the equation:

\[
\left(\log_2(5) + \log_2(x)\right) - \frac{1}{2}(1 + \log_2(x)) = 3
\]

### Step 3: Solve for \(\log_2(x)\)

Distribute the \(\frac{1}{2}\) in the second term:

\[
\log_2(5) + \log_2(x) - \frac{1}{2} - \frac{1}{2}\log_2(x) = 3
\]

Combine like terms:

\[
\log_2(5) + \left(1 - \frac{1}{2}\right)\log_2(x) - \frac{1}{2} = 3
\]

Simplify further:

\[
\log_2(5) + \frac{1}{2} \log_2(x) - \frac{1}{2} = 3
\]

Now, isolate \(\log_2(x)\):

\[
\frac{1}{2} \log_2(x) = 3 + \frac{1}{2} - \log_2(5)
\]

\[
\frac{1}{2} \log_2(x) = \frac{7}{2} - \log_2(5)
\]

Multiply both sides by 2:

\[
\log_2(x) = 7 - 2\log_2(5)
\]

### Step 4: Solve for \(x\)

Exponentiate both sides to remove the logarithm:

\[
x = 2^{7 - 2\log_2(5)} = 2^7 \cdot 2^{-2\log_2(5)}
\]

Since \(2^{\log_2(a)} = a\), this becomes:

\[
x = 128 \cdot 5^{-2}
\]

\[
x = 128 \cdot \frac{1}{25} = \frac{128}{25}
\]

Thus, the solution is:

\[
x = \frac{128}{25}
\]

Let me know if you'd like further clarification on any part of this process!

### Related Questions:
1. How do you change the base of a logarithm?
2. What is the logarithmic identity for \(\log_b(ab)\)?
3. How do you simplify logarithmic expressions with coefficients?
4. How can you solve logarithmic equations using properties of exponents?
5. What is the purpose of the change of base formula?

### Tip:
When dealing with logarithms of different bases, converting everything to a common base simplifies the process tremendously.

Solve for x given that log2(5x) - log4(2x) = 3

Learn how to solve for x in the equation log2(5x) - log4(2x) = 3. This detailed solution covers the change of base formula, properties of logarithms, and exponentiation rules. Perfect for high school students in Grades 10-12.

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