Find value of x X[log3-log10]=(1-x) long20

The equation you provided is:

\[
x \cdot (\log{3} - \log{10}) = (1 - x) \cdot \log{20}
\]

To solve for \( x \), follow these steps:

### Step 1: Expand the logarithms
We can simplify the equation by recalling that:

\[
\log{a} - \log{b} = \log{\left(\frac{a}{b}\right)}
\]

Thus, the equation becomes:

\[
x \cdot \log{\left(\frac{3}{10}\right)} = (1 - x) \cdot \log{20}
\]

### Step 2: Isolate \( x \)
Now, expand the equation to distribute \( x \) and \( 1-x \):

\[
x \cdot \log{\left(\frac{3}{10}\right)} = \log{20} - x \cdot \log{20}
\]

Move all terms involving \( x \) to one side:

\[
x \cdot \log{\left(\frac{3}{10}\right)} + x \cdot \log{20} = \log{20}
\]

Factor out \( x \):

\[
x \cdot \left(\log{\left(\frac{3}{10}\right)} + \log{20}\right) = \log{20}
\]

### Step 3: Solve for \( x \)
Finally, solve for \( x \):

\[
x = \frac{\log{20}}{\log{\left(\frac{3}{10}\right)} + \log{20}}
\]

### Step 4: Simplify the equation
Recognize that \(\log{\left(\frac{3}{10}\right)}\) is \(\log{3} - \log{10}\). Substitute this back into the equation:

\[
x = \frac{\log{20}}{\log{3} - \log{10} + \log{20}}
\]

This expression gives you the value of \( x \).

Would you like me to calculate the exact numerical value of \( x \), or do you have any questions about the steps? 

---

### Follow-up Questions:
1. How do logarithmic properties help simplify complex equations?
2. What are the key differences between natural logarithms and common logarithms?
3. How can logarithms be applied in solving exponential growth problems?
4. Can you explain the relationship between logarithms and exponents?
5. What are the practical applications of logarithms in real-world scenarios?

### Tip:
Always remember that logarithmic identities are powerful tools for simplifying complex expressions, especially in equations involving products or ratios.

Learn how to solve the logarithmic equation involving logarithm properties and equations. This problem covers key concepts in logarithms and their applications in equations. Suitable for advanced high school students and anyone interested in logarithmic equations.

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