Analyzing A Solution:Explain The Error In The Solution Below. What Additional Step Needs To Be Completed?$\[ \begin{aligned} \log X - \log_5 3 & = 2 \log_5 3 \\ \log X & = 3 \log_5 3 \\ \log X & = \log_5 3^3 \\ x & = 27 \end{aligned} \\]The

Feb 26, 2025 by ADMIN 241 views

Analyzing a Solution: Identifying the Error and Additional Step

Introduction

In mathematics, solving logarithmic equations requires a deep understanding of the properties and rules governing logarithms. However, even with a solid grasp of these concepts, errors can still occur. In this analysis, we will examine a given solution to a logarithmic equation and identify the error in the solution. We will also determine the additional step needed to complete the solution.

The Given Solution

The given solution is as follows:

{ \begin{aligned} \log x - \log_5 3 & = 2 \log_5 3 \\ \log x & = 3 \log_5 3 \\ \log x & = \log_5 3^3 \\ x & = 27 \end{aligned} \}

Identifying the Error

Upon examining the solution, we notice that the error occurs in the second step. The equation $\log x = 3 \log_5 3$ is incorrect. This is because the property of logarithms used here is not applicable. The correct property to use is the product rule of logarithms, which states that $\log a - \log b = \log \frac{a}{b}$ . However, in this case, we have $\log x - \log_5 3 = 2 \log_5 3$ , which cannot be simplified using the product rule.

Correcting the Error

To correct the error, we need to apply the correct property of logarithms. We can start by using the quotient rule of logarithms, which states that $\log a - \log b = \log \frac{a}{b}$ . Applying this rule to the given equation, we get:

$\log x - \log_5 3 = 2 \log_5 3$

$\log x = \log_5 3^2 + \log_5 3$

$\log x = \log_5 3^3$

The Correct Solution

Now that we have corrected the error, we can proceed with the solution. The correct solution is as follows:

{ \begin{aligned} \log x - \log_5 3 & = 2 \log_5 3 \\ \log x & = \log_5 3^2 + \log_5 3 \\ \log x & = \log_5 3^3 \\ x & = 5^3 \\ x & = 125 \end{aligned} \}

Conclusion

In conclusion, the error in the given solution was in the second step, where the incorrect property of logarithms was used. By applying the correct property, we were able to correct the error and obtain the correct solution. The additional step needed to complete the solution was to apply the quotient rule of logarithms and simplify the equation.

Additional Step: Applying the Quotient Rule of Logarithms

The additional step needed to complete the solution is to apply the quotient rule of logarithms. This rule states that $\log a - \log b = \log \frac{a}{b}$ . By applying this rule, we can simplify the equation and obtain the correct solution.

Importance of Understanding Logarithmic Properties

Understanding logarithmic properties is crucial in solving logarithmic equations. The properties of logarithms provide a set of rules that can be used to simplify and solve logarithmic equations. By applying these properties correctly, we can obtain the correct solution to a logarithmic equation.

Common Errors in Logarithmic Equations

Common errors in logarithmic equations include:

Using the incorrect property of logarithms
Not applying the correct property of logarithms
Not simplifying the equation correctly

Conclusion

Final Thoughts

Solving logarithmic equations requires a deep understanding of the properties and rules governing logarithms. By understanding these properties and applying them correctly, we can obtain the correct solution to a logarithmic equation. It is essential to be aware of common errors in logarithmic equations and to take the necessary steps to correct them.

References

[1] "Logarithmic Equations" by Math Open Reference
[2] "Properties of Logarithms" by Math Is Fun
[3] "Solving Logarithmic Equations" by Khan Academy

Introduction

Logarithmic equations can be challenging to solve, but with a solid understanding of the properties and rules governing logarithms, you can tackle even the most complex equations. In this Q&A guide, we will cover common questions and topics related to logarithmic equations, providing you with a comprehensive understanding of this important mathematical concept.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. Logarithmic equations can be solved using various properties and rules of logarithms.

Q: What are the common properties of logarithms?

A: The common properties of logarithms include:

Product Rule: $\log a + \log b = \log (ab)$
Quotient Rule: $\log a - \log b = \log \frac{a}{b}$
Power Rule: $\log a^b = b \log a$
Change of Base Rule: $\log_a b = \frac{\log_c b}{\log_c a}$

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, follow these steps:

Isolate the logarithm: Move all terms except the logarithm to one side of the equation.
Apply the properties of logarithms: Use the product rule, quotient rule, power rule, or change of base rule to simplify the equation.
Exponentiate both sides: Raise both sides of the equation to the power of the base of the logarithm.
Solve for the variable: Simplify the resulting equation to solve for the variable.

Q: What is the difference between a logarithmic equation and an exponential equation?

A: A logarithmic equation involves a logarithm, which is the inverse operation of exponentiation. An exponential equation, on the other hand, involves an exponent, which is the inverse operation of a logarithm. For example:

Logarithmic Equation: $\log x = 2$
Exponential Equation: $x^2 = 4$

Q: How do I determine the base of a logarithm?

A: The base of a logarithm is the number that is raised to a power to produce the argument of the logarithm. For example:

Logarithm with base 2: $\log_2 x$
Logarithm with base 10: $\log_{10} x$

Q: What is the relationship between logarithms and exponents?

A: Logarithms and exponents are inverse operations. This means that if $a^b = c$ , then $\log_a c = b$ . For example:

Exponential Equation: $2^3 = 8$
Logarithmic Equation: $\log_2 8 = 3$

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, follow these steps:

Determine the base: Identify the base of the logarithm.
Determine the argument: Identify the argument of the logarithm.
Apply the properties of logarithms: Use the product rule, quotient rule, power rule, or change of base rule to simplify the expression.
Evaluate the expression: Simplify the resulting expression to obtain the final answer.

Q: What are some common mistakes to avoid when solving logarithmic equations?

A: Some common mistakes to avoid when solving logarithmic equations include:

Using the incorrect property of logarithms
Not applying the correct property of logarithms
Not simplifying the equation correctly
Not checking the domain of the logarithm

Conclusion

Logarithmic equations can be challenging to solve, but with a solid understanding of the properties and rules governing logarithms, you can tackle even the most complex equations. By following the steps outlined in this Q&A guide, you can develop a comprehensive understanding of logarithmic equations and become proficient in solving them.

Final Thoughts

Solving logarithmic equations requires a deep understanding of the properties and rules governing logarithms. By understanding these properties and applying them correctly, you can obtain the correct solution to a logarithmic equation. It is essential to be aware of common errors in logarithmic equations and to take the necessary steps to correct them.

References

[1] "Logarithmic Equations" by Math Open Reference
[2] "Properties of Logarithms" by Math Is Fun
[3] "Solving Logarithmic Equations" by Khan Academy