To solve the equation \( 3 \cdot \log_2(x) - 2 \cdot \log(5x) = 2 \), we need to work through the following steps:

### Step 1: Convert the logarithms to a common base
Given that the equation mixes base-2 logarithms with base-10 logarithms, it’s usually easiest to convert everything to the same base. Let's convert \(\log_2(x)\) to base 10 using the change of base formula:

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

So, the equation becomes:

\[
3 \cdot \frac{\log(x)}{\log(2)} - 2 \cdot \log(5x) = 2
\]

### Step 2: Simplify the equation
Distribute and simplify the terms:

\[
\frac{3 \cdot \log(x)}{\log(2)} - 2 \cdot \log(5x) = 2
\]

### Step 3: Express \(\log(5x)\)
Recall that:

\[
\log(5x) = \log(5) + \log(x)
\]

Substitute this back into the equation:

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

Expanding:

\[
\frac{3 \cdot \log(x)}{\log(2)} - 2 \cdot \log(5) - 2 \cdot \log(x) = 2
\]

### Step 4: Combine like terms
Group the \(\log(x)\) terms together:

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

### Step 5: Solve for \(\log(x)\)
Now isolate \(\log(x)\):

\[
\log(x) = \frac{2 + 2 \cdot \log(5)}{\frac{3}{\log(2)} - 2}
\]

### Step 6: Calculate the value
Substitute \(\log(2) \approx 0.3010\) and \(\log(5) \approx 0.6990\):

\[
\log(x) = \frac{2 + 2 \cdot 0.6990}{\frac{3}{0.3010} - 2}
\]

Calculate:

\[
\log(x) = \frac{2 + 1.398}{9.9668 - 2}
\]

\[
\log(x) = \frac{3.398}{7.9668} \approx 0.4267
\]

Finally, solve for \(x\):

\[
x = 10^{0.4267} \approx 2.67
\]

### Final Answer:
The solution is approximately \(x \approx 2.67\).

---

Do you want any details or have any questions?

Here are 5 related questions for further practice:
1. Solve \(2 \cdot \log_3(x) + 3 \cdot \log_2(x) = 5\).
2. Solve \( \log_4(x) + 2 \cdot \log(x) = 3 \).
3. What is the value of \( x \) if \( 5 \cdot \log(x) - \log_5(x^2) = 1 \)?
4. Solve for \( x \) in the equation \( \log(x^2) + \log(x) = 3 \).
5. Find the value of \( x \) in the equation \( 4 \cdot \log_5(x) - \log_5(x^3) = 1 \).

**Tip:** Always consider converting logarithms to the same base when solving equations that mix different logarithmic bases.

Discover how to solve the logarithmic equation 3*log_2(x) - 2*log(5x) = 2 using the change of base formula and properties of logarithms. This detailed step-by-step solution is ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation