Simplify The Expression:\[$\log (x+5) - \log 4 - \log (x+2)\$\]

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Introduction

Logarithmic expressions can be complex and challenging to simplify, but with the right techniques and strategies, they can be broken down into manageable parts. In this article, we will explore the process of simplifying a logarithmic expression involving logarithms with different bases and arguments.

Understanding Logarithmic Properties

Before we dive into simplifying the given expression, it's essential to understand the properties of logarithms. The logarithm of a number is the exponent to which a base must be raised to produce that number. For example, the logarithm of 100 to the base 10 is 2, because 10^2 = 100.

There are several key properties of logarithms that we will use to simplify the expression:

  • Product Property: log(a * b) = log(a) + log(b)
  • Quotient Property: log(a / b) = log(a) - log(b)
  • Power Property: log(a^b) = b * log(a)

Simplifying the Given Expression

The given expression is:

log⁑(x+5)βˆ’log⁑4βˆ’log⁑(x+2)\log (x+5) - \log 4 - \log (x+2)

To simplify this expression, we can use the quotient property of logarithms, which states that log(a / b) = log(a) - log(b). We can rewrite the expression as:

log⁑((x+5)/(x+2))βˆ’log⁑4\log ((x+5) / (x+2)) - \log 4

Applying the Quotient Property

Now, we can apply the quotient property to simplify the expression further:

log⁑((x+5)/(x+2))βˆ’log⁑4=log⁑((x+5)/(x+2))βˆ’log⁑4\log ((x+5) / (x+2)) - \log 4 = \log ((x+5) / (x+2)) - \log 4

Using the Product Property

We can use the product property to rewrite the expression as:

log⁑((x+5)/(x+2))βˆ’log⁑4=log⁑((x+5)/(x+2))βˆ’log⁑(22)\log ((x+5) / (x+2)) - \log 4 = \log ((x+5) / (x+2)) - \log (2^2)

Simplifying the Expression Further

Now, we can simplify the expression further by using the power property of logarithms, which states that log(a^b) = b * log(a). We can rewrite the expression as:

log⁑((x+5)/(x+2))βˆ’log⁑(22)=log⁑((x+5)/(x+2))βˆ’2log⁑2\log ((x+5) / (x+2)) - \log (2^2) = \log ((x+5) / (x+2)) - 2 \log 2

Using the Quotient Property Again

We can use the quotient property again to simplify the expression further:

log⁑((x+5)/(x+2))βˆ’2log⁑2=log⁑((x+5)/(x+2))βˆ’log⁑(22)\log ((x+5) / (x+2)) - 2 \log 2 = \log ((x+5) / (x+2)) - \log (2^2)

Simplifying the Expression to Its Final Form

Now, we can simplify the expression to its final form by combining the logarithms:

log⁑((x+5)/(x+2))βˆ’log⁑(22)=log⁑((x+5)/(x+2))βˆ’log⁑4\log ((x+5) / (x+2)) - \log (2^2) = \log ((x+5) / (x+2)) - \log 4

Conclusion

In this article, we explored the process of simplifying a logarithmic expression involving logarithms with different bases and arguments. We used the quotient property, product property, and power property of logarithms to simplify the expression step by step. The final simplified expression is:

log⁑((x+5)/(x+2))βˆ’log⁑4\log ((x+5) / (x+2)) - \log 4

This expression can be further simplified by combining the logarithms, but it is already in its simplest form.

Final Answer

The final answer is log⁑((x+5)/(x+2))βˆ’log⁑4\boxed{\log ((x+5) / (x+2)) - \log 4}.

Additional Tips and Tricks

  • When simplifying logarithmic expressions, it's essential to use the correct properties of logarithms.
  • Always check your work by plugging in values for the variables to ensure that the expression is simplified correctly.
  • Practice simplifying logarithmic expressions to become more comfortable with the properties and techniques used in this article.

Common Mistakes to Avoid

  • Don't forget to use the correct properties of logarithms when simplifying expressions.
  • Be careful when combining logarithms, as it's easy to make mistakes.
  • Always check your work by plugging in values for the variables to ensure that the expression is simplified correctly.

Real-World Applications

Logarithmic expressions are used in a wide range of real-world applications, including:

  • Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
  • Science: Logarithmic expressions are used to calculate the pH of a solution and the concentration of a substance.
  • Engineering: Logarithmic expressions are used to calculate the power consumption of a device and the efficiency of a system.

Conclusion

In conclusion, simplifying logarithmic expressions is a crucial skill in mathematics and has many real-world applications. By understanding the properties of logarithms and using the correct techniques, you can simplify even the most complex expressions. Remember to always check your work and practice simplifying logarithmic expressions to become more comfortable with the properties and techniques used in this article.

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Introduction

In our previous article, we explored the process of simplifying a logarithmic expression involving logarithms with different bases and arguments. We used the quotient property, product property, and power property of logarithms to simplify the expression step by step. In this article, we will answer some frequently asked questions about simplifying logarithmic expressions.

Q&A

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

A: A logarithmic expression is an expression that involves a logarithm, which is the inverse of an exponential expression. An exponential expression is an expression that involves an exponent, which is a power to which a base is raised.

Q: How do I simplify a logarithmic expression with a negative exponent?

A: To simplify a logarithmic expression with a negative exponent, you can use the property of logarithms that states log(a^(-b)) = -b * log(a). This means that you can rewrite the expression as a negative multiple of the logarithm of the base.

Q: Can I simplify a logarithmic expression with a fraction as the argument?

A: Yes, you can simplify a logarithmic expression with a fraction as the argument by using the quotient property of logarithms, which states that log(a / b) = log(a) - log(b).

Q: How do I simplify a logarithmic expression with a product or quotient of logarithms?

A: To simplify a logarithmic expression with a product or quotient of logarithms, you can use the product property and quotient property of logarithms, which state that log(a * b) = log(a) + log(b) and log(a / b) = log(a) - log(b).

Q: Can I simplify a logarithmic expression with a logarithm of a logarithm?

A: Yes, you can simplify a logarithmic expression with a logarithm of a logarithm by using the power property of logarithms, which states that log(a^b) = b * log(a).

Q: How do I check my work when simplifying a logarithmic expression?

A: To check your work when simplifying a logarithmic expression, you can plug in values for the variables and see if the expression is simplified correctly. You can also use a calculator to check your work.

Common Mistakes to Avoid

  • Forgetting to use the correct properties of logarithms: Make sure to use the correct properties of logarithms when simplifying expressions.
  • Not checking your work: Always check your work by plugging in values for the variables to ensure that the expression is simplified correctly.
  • Not using the correct order of operations: Make sure to use the correct order of operations when simplifying expressions.

Real-World Applications

Logarithmic expressions are used in a wide range of real-world applications, including:

  • Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
  • Science: Logarithmic expressions are used to calculate the pH of a solution and the concentration of a substance.
  • Engineering: Logarithmic expressions are used to calculate the power consumption of a device and the efficiency of a system.

Conclusion

In conclusion, simplifying logarithmic expressions is a crucial skill in mathematics and has many real-world applications. By understanding the properties of logarithms and using the correct techniques, you can simplify even the most complex expressions. Remember to always check your work and practice simplifying logarithmic expressions to become more comfortable with the properties and techniques used in this article.

Final Tips and Tricks

  • Practice, practice, practice: The more you practice simplifying logarithmic expressions, the more comfortable you will become with the properties and techniques used in this article.
  • Use a calculator to check your work: A calculator can be a useful tool for checking your work when simplifying logarithmic expressions.
  • Read the problem carefully: Make sure to read the problem carefully and understand what is being asked before simplifying the expression.

Common Logarithmic Expressions

Here are some common logarithmic expressions that you may encounter:

  • log(x): This is a logarithmic expression with a base of 10.
  • log(x) / log(y): This is a logarithmic expression with a base of 10 and a quotient of logarithms.
  • log(x^y): This is a logarithmic expression with a base of 10 and a power of logarithms.
  • log(x + y): This is a logarithmic expression with a base of 10 and a product of logarithms.

Logarithmic Identities

Here are some common logarithmic identities that you may encounter:

  • log(a) + log(b) = log(a * b): This is the product property of logarithms.
  • log(a) - log(b) = log(a / b): This is the quotient property of logarithms.
  • log(a^b) = b * log(a): This is the power property of logarithms.
  • log(a) = log(b) if and only if a = b: This is the one-to-one property of logarithms.