Solve The Equation:$\[ \frac{\frac{x}{2} \log E}{\log E} = \frac{\log \frac{1}{10}}{\log E} \\]

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Introduction

In this article, we will delve into solving a complex equation involving logarithms. The equation is given as:

x2log⁡Elog⁡E=log⁡110log⁡E\frac{\frac{x}{2} \log E}{\log E} = \frac{\log \frac{1}{10}}{\log E}

We will break down the solution step by step, using mathematical concepts and techniques to simplify the equation and arrive at the final answer.

Understanding the Equation

The given equation involves logarithms and fractions. To start solving it, we need to understand the properties of logarithms and how to manipulate them.

Logarithmic Properties

The logarithm of a number is the exponent to which a base number must be raised to produce that number. In this case, we are dealing with the natural logarithm (base e) and the common logarithm (base 10).

  • Logarithmic Identity: log⁡ba=log⁡calog⁡cb\log_b a = \frac{\log_c a}{\log_c b}, where bb and cc are positive numbers not equal to 1.
  • Logarithmic Power Rule: log⁡b(ac)=clog⁡ba\log_b (a^c) = c \log_b a

Simplifying the Equation

To simplify the equation, we can start by canceling out the common factors.

x2log⁡Elog⁡E=log⁡110log⁡E\frac{\frac{x}{2} \log E}{\log E} = \frac{\log \frac{1}{10}}{\log E}

Since log⁡E\log E appears in both the numerator and denominator on the left-hand side, we can cancel it out.

x2=log⁡110log⁡E\frac{x}{2} = \frac{\log \frac{1}{10}}{\log E}

Applying Logarithmic Properties

Now, we can apply the logarithmic power rule to simplify the right-hand side of the equation.

log⁡110=log⁡(10−1)=−log⁡10\log \frac{1}{10} = \log (10^{-1}) = -\log 10

Substituting this back into the equation, we get:

x2=−log⁡10log⁡E\frac{x}{2} = \frac{-\log 10}{\log E}

Using the Change of Base Formula

To simplify the equation further, we can use the change of base formula to express the logarithms in terms of a common base.

log⁡E=log⁡Elog⁡E=log⁡Elog⁡10\log E = \frac{\log E}{\log E} = \frac{\log E}{\log 10}

Substituting this back into the equation, we get:

x2=−log⁡10log⁡Elog⁡10\frac{x}{2} = \frac{-\log 10}{\frac{\log E}{\log 10}}

Simplifying the Fraction

To simplify the fraction, we can multiply the numerator and denominator by log⁡10\log 10.

x2=−log⁡10⋅log⁡10log⁡E\frac{x}{2} = \frac{-\log 10 \cdot \log 10}{\log E}

Canceling Out Common Factors

Since log⁡10\log 10 appears in both the numerator and denominator, we can cancel it out.

x2=−log⁡Elog⁡E\frac{x}{2} = \frac{-\log E}{\log E}

Solving for x

Now, we can solve for xx by multiplying both sides of the equation by 2.

x=−2x = -2

Conclusion

In this article, we solved a complex equation involving logarithms using mathematical concepts and techniques. We applied logarithmic properties, simplified fractions, and used the change of base formula to arrive at the final answer.

Key Takeaways

  • Logarithmic properties can be used to simplify complex equations.
  • The change of base formula can be used to express logarithms in terms of a common base.
  • Simplifying fractions and canceling out common factors can help to arrive at the final answer.

By following these steps and using mathematical concepts and techniques, we can solve complex equations involving logarithms and arrive at the final answer.

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Introduction

In our previous article, we solved a complex equation involving logarithms using mathematical concepts and techniques. In this article, we will provide a Q&A guide to help you understand the solution and answer any questions you may have.

Q: What is the equation we solved?

A: The equation we solved is:

x2log⁡Elog⁡E=log⁡110log⁡E\frac{\frac{x}{2} \log E}{\log E} = \frac{\log \frac{1}{10}}{\log E}

Q: What are logarithms?

A: Logarithms are the exponents to which a base number must be raised to produce a given number. In this case, we are dealing with the natural logarithm (base e) and the common logarithm (base 10).

Q: What is the change of base formula?

A: The change of base formula is a mathematical formula that allows us to express logarithms in terms of a common base. It is given by:

log⁡ba=log⁡calog⁡cb\log_b a = \frac{\log_c a}{\log_c b}

Q: How did we simplify the equation?

A: We simplified the equation by canceling out common factors, applying logarithmic properties, and using the change of base formula.

Q: What is the final answer?

A: The final answer is x=−2x = -2.

Q: Why did we use the change of base formula?

A: We used the change of base formula to express the logarithms in terms of a common base, which allowed us to simplify the equation.

Q: What are some common logarithmic properties?

A: Some common logarithmic properties include:

  • Logarithmic Identity: log⁡ba=log⁡calog⁡cb\log_b a = \frac{\log_c a}{\log_c b}
  • Logarithmic Power Rule: log⁡b(ac)=clog⁡ba\log_b (a^c) = c \log_b a

Q: How can I apply logarithmic properties to simplify equations?

A: You can apply logarithmic properties to simplify equations by canceling out common factors, using the change of base formula, and applying the logarithmic power rule.

Q: What are some tips for solving complex equations involving logarithms?

A: Some tips for solving complex equations involving logarithms include:

  • Start by simplifying the equation: Use logarithmic properties and the change of base formula to simplify the equation.
  • Use algebraic techniques: Use algebraic techniques such as factoring and canceling out common factors to simplify the equation.
  • Check your work: Check your work by plugging the final answer back into the original equation.

Conclusion

In this article, we provided a Q&A guide to help you understand the solution to the complex equation involving logarithms. We answered common questions and provided tips for solving complex equations involving logarithms.

Key Takeaways

  • Logarithmic properties can be used to simplify complex equations.
  • The change of base formula can be used to express logarithms in terms of a common base.
  • Algebraic techniques such as factoring and canceling out common factors can be used to simplify the equation.

By following these tips and using logarithmic properties and algebraic techniques, you can solve complex equations involving logarithms and arrive at the final answer.