Math Problem Statement

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

Solution

The equation you provided is:

x(log3log10)=(1x)log20x \cdot (\log{3} - \log{10}) = (1 - x) \cdot \log{20}

To solve for xx, follow these steps:

Step 1: Expand the logarithms

We can simplify the equation by recalling that:

logalogb=log(ab)\log{a} - \log{b} = \log{\left(\frac{a}{b}\right)}

Thus, the equation becomes:

xlog(310)=(1x)log20x \cdot \log{\left(\frac{3}{10}\right)} = (1 - x) \cdot \log{20}

Step 2: Isolate xx

Now, expand the equation to distribute xx and 1x1-x:

xlog(310)=log20xlog20x \cdot \log{\left(\frac{3}{10}\right)} = \log{20} - x \cdot \log{20}

Move all terms involving xx to one side:

xlog(310)+xlog20=log20x \cdot \log{\left(\frac{3}{10}\right)} + x \cdot \log{20} = \log{20}

Factor out xx:

x(log(310)+log20)=log20x \cdot \left(\log{\left(\frac{3}{10}\right)} + \log{20}\right) = \log{20}

Step 3: Solve for xx

Finally, solve for xx:

x=log20log(310)+log20x = \frac{\log{20}}{\log{\left(\frac{3}{10}\right)} + \log{20}}

Step 4: Simplify the equation

Recognize that log(310)\log{\left(\frac{3}{10}\right)} is log3log10\log{3} - \log{10}. Substitute this back into the equation:

x=log20log3log10+log20x = \frac{\log{20}}{\log{3} - \log{10} + \log{20}}

This expression gives you the value of xx.

Would you like me to calculate the exact numerical value of xx, 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.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Equations
Solving Equations

Formulas

Logarithm properties
Exponential equations

Theorems

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Suitable Grade Level

Advanced High School