Which Equation Is Equivalent To $\log 5 X^3 - \log X^2 = 2$?A. $10^{\log 5 X^3} = 10^2$B. $10^{\log \frac{5 X^3}{x^2}} = 10^2$C. $10^{\log \frac{5 X^3 + X^2}{x^7}} = 10^2$

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Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be broken down into manageable steps. In this article, we will explore how to solve the equation log⁡5x3−log⁡x2=2\log 5 x^3 - \log x^2 = 2 and determine which of the given options is equivalent to it.

Understanding Logarithmic Properties

Before we dive into solving the equation, it's essential to understand the properties of logarithms. The two main properties we will use in this article are:

  • Product Property: log⁡(ab)=log⁡a+log⁡b\log (ab) = \log a + \log b
  • Quotient Property: log⁡(ab)=log⁡a−log⁡b\log \left(\frac{a}{b}\right) = \log a - \log b

Solving the Equation

To solve the equation log⁡5x3−log⁡x2=2\log 5 x^3 - \log x^2 = 2, we can start by using the Quotient Property to combine the two logarithmic terms.

log⁡5x3−log⁡x2=log⁡(5x3x2)\log 5 x^3 - \log x^2 = \log \left(\frac{5 x^3}{x^2}\right)

Now, we can simplify the expression inside the logarithm.

log⁡(5x3x2)=log⁡(5x)\log \left(\frac{5 x^3}{x^2}\right) = \log (5 x)

The equation now becomes:

log⁡(5x)=2\log (5 x) = 2

To solve for xx, we can use the definition of a logarithm, which states that log⁡a=b\log a = b is equivalent to a=10ba = 10^b.

5x=1025 x = 10^2

Now, we can solve for xx.

x=1025x = \frac{10^2}{5}

x=20x = 20

Evaluating the Options

Now that we have solved the equation, we can evaluate the given options to determine which one is equivalent to it.

Option A

10log⁡5x3=10210^{\log 5 x^3} = 10^2

Using the definition of a logarithm, we can rewrite the left-hand side of the equation as:

5x3=1025 x^3 = 10^2

This is not equivalent to the original equation, so we can eliminate option A.

Option B

10log⁡5x3x2=10210^{\log \frac{5 x^3}{x^2}} = 10^2

Using the definition of a logarithm, we can rewrite the left-hand side of the equation as:

5x3x2=102\frac{5 x^3}{x^2} = 10^2

Simplifying the expression, we get:

5x=1025 x = 10^2

This is equivalent to the original equation, so we can conclude that option B is the correct answer.

Option C

10log⁡5x3+x2x7=10210^{\log \frac{5 x^3 + x^2}{x^7}} = 10^2

Using the definition of a logarithm, we can rewrite the left-hand side of the equation as:

5x3+x2x7=102\frac{5 x^3 + x^2}{x^7} = 10^2

Simplifying the expression, we get:

5x3+x2x7=100\frac{5 x^3 + x^2}{x^7} = 100

This is not equivalent to the original equation, so we can eliminate option C.

Conclusion

In this article, we solved the equation log⁡5x3−log⁡x2=2\log 5 x^3 - \log x^2 = 2 and determined that option B is the correct answer. We used the properties of logarithms to simplify the equation and then used the definition of a logarithm to solve for xx. This approach can be applied to a wide range of logarithmic equations, making it an essential tool for anyone working with logarithms.

Final Answer

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

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. An exponential equation, on the other hand, is an equation that involves an exponent, which is a power to which a number is raised.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the properties of logarithms to simplify the equation and then use the definition of a logarithm to solve for the variable. The two main properties of logarithms are the product property and the quotient property.

Q: What is the product property of logarithms?

A: The product property of logarithms states that log⁡(ab)=log⁡a+log⁡b\log (ab) = \log a + \log b. This means that the logarithm of a product is equal to the sum of the logarithms of the factors.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that log⁡(ab)=log⁡a−log⁡b\log \left(\frac{a}{b}\right) = \log a - \log b. This means that the logarithm of a quotient is equal to the difference of the logarithms of the dividend and the divisor.

Q: How do I use the definition of a logarithm to solve for a variable?

A: To use the definition of a logarithm to solve for a variable, you can rewrite the equation in exponential form. For example, if you have the equation log⁡x=2\log x = 2, you can rewrite it as x=102x = 10^2.

Q: What is the difference between a base-10 logarithm and a base-e logarithm?

A: A base-10 logarithm is a logarithm with a base of 10, while a base-e logarithm is a logarithm with a base of e (approximately 2.718). The two types of logarithms are related by the equation log⁡10x=ln⁡xln⁡10\log_{10} x = \frac{\ln x}{\ln 10}.

Q: How do I convert a base-10 logarithm to a base-e logarithm?

A: To convert a base-10 logarithm to a base-e logarithm, you can use the equation log⁡10x=ln⁡xln⁡10\log_{10} x = \frac{\ln x}{\ln 10}. This means that you can convert a base-10 logarithm to a base-e logarithm by dividing the result by the natural logarithm of 10.

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

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

  • Forgetting to use the properties of logarithms to simplify the equation
  • Not using the definition of a logarithm to solve for the variable
  • Making errors when simplifying the equation
  • Not checking the solution to make sure it is valid

Q: How do I check the solution to a logarithmic equation?

A: To check the solution to a logarithmic equation, you can plug the solution back into the original equation and make sure it is true. You can also use a calculator to check the solution.

Q: What are some real-world applications of logarithmic equations?

A: Logarithmic equations have many real-world applications, including:

  • Finance: Logarithmic equations are used to calculate interest rates and investment returns.
  • Science: Logarithmic equations are used to model population growth and decay.
  • Engineering: Logarithmic equations are used to design and optimize systems.
  • Computer Science: Logarithmic equations are used to analyze and optimize algorithms.

Conclusion

In this article, we have answered some frequently asked questions about logarithmic equations. We have covered topics such as the difference between a logarithmic equation and an exponential equation, how to solve a logarithmic equation, and common mistakes to avoid when solving logarithmic equations. We have also discussed real-world applications of logarithmic equations and how to check the solution to a logarithmic equation.