Fill In The Missing Values To Make The Equations True.(a) $\log _4 5+\log _4 11=\log _4 \square$(b) $\log _2 7-\log _2 \square=\log _2 \frac{7}{5}$(c) $\log _3 \frac{1}{25}=\square \log _3 5$

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Introduction

In this article, we will explore three logarithmic equations and fill in the missing values to make them true. Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. We will use the properties of logarithms to simplify the equations and find the missing values.

Equation (a) - Logarithmic Addition

The first equation is log⁡45+log⁡411=log⁡4□\log _4 5+\log _4 11=\log _4 \square. This equation involves the addition of two logarithms with the same base. We can use the property of logarithmic addition to simplify the equation.

Logarithmic Addition Property

The logarithmic addition property states that log⁡ax+log⁡ay=log⁡a(xy)\log _a x + \log _a y = \log _a (xy). This property allows us to combine the two logarithms on the left-hand side of the equation into a single logarithm.

Applying the Logarithmic Addition Property

Using the logarithmic addition property, we can rewrite the equation as log⁡4(5⋅11)=log⁡4□\log _4 (5 \cdot 11) = \log _4 \square. Simplifying the expression inside the logarithm, we get log⁡455=log⁡4□\log _4 55 = \log _4 \square.

Finding the Missing Value

Since the bases of the logarithms are the same, we can equate the expressions inside the logarithms. Therefore, we have 55=□55 = \square. The missing value is 55\boxed{55}.

Equation (b) - Logarithmic Subtraction

The second equation is log⁡27−log⁡2□=log⁡275\log _2 7-\log _2 \square=\log _2 \frac{7}{5}. This equation involves the subtraction of two logarithms with the same base. We can use the property of logarithmic subtraction to simplify the equation.

Logarithmic Subtraction Property

The logarithmic subtraction property states that log⁡ax−log⁡ay=log⁡a(xy)\log _a x - \log _a y = \log _a \left(\frac{x}{y}\right). This property allows us to combine the two logarithms on the left-hand side of the equation into a single logarithm.

Applying the Logarithmic Subtraction Property

Using the logarithmic subtraction property, we can rewrite the equation as log⁡2(7□)=log⁡275\log _2 \left(\frac{7}{\square}\right) = \log _2 \frac{7}{5}. Since the bases of the logarithms are the same, we can equate the expressions inside the logarithms.

Finding the Missing Value

Equating the expressions inside the logarithms, we get 7□=75\frac{7}{\square} = \frac{7}{5}. Solving for the missing value, we find that □=5\square = \boxed{5}.

Equation (c) - Logarithmic Multiplication

The third equation is log⁡3125=□log⁡35\log _3 \frac{1}{25}=\square \log _3 5. This equation involves the multiplication of a logarithm by a constant. We can use the property of logarithmic multiplication to simplify the equation.

Logarithmic Multiplication Property

The logarithmic multiplication property states that alog⁡bx=log⁡bxaa \log _b x = \log _b x^a. This property allows us to combine the constant with the logarithm.

Applying the Logarithmic Multiplication Property

Using the logarithmic multiplication property, we can rewrite the equation as log⁡3(125)□=log⁡35□\log _3 \left(\frac{1}{25}\right)^{\square} = \log _3 5^{\square}. Since the bases of the logarithms are the same, we can equate the expressions inside the logarithms.

Finding the Missing Value

Equating the expressions inside the logarithms, we get (125)□=5□\left(\frac{1}{25}\right)^{\square} = 5^{\square}. Solving for the missing value, we find that □=−2\square = \boxed{-2}.

Conclusion

In this article, we have filled in the missing values to make three logarithmic equations true. We have used the properties of logarithms to simplify the equations and find the missing values. The missing values are 55\boxed{55}, 5\boxed{5}, and −2\boxed{-2}, respectively. These values satisfy the equations and demonstrate the power of logarithmic properties in solving mathematical problems.

References

  • [1] "Logarithmic Properties" by Math Open Reference. Retrieved February 2023.
  • [2] "Logarithmic Equations" by Khan Academy. Retrieved February 2023.
  • [3] "Properties of Logarithms" by Wolfram MathWorld. Retrieved February 2023.

Introduction

In the previous article, we explored three logarithmic equations and filled in the missing values to make them true. Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will answer some frequently asked questions about logarithmic equations.

Q&A

Q: What is the difference between logarithmic addition and logarithmic subtraction?

A: Logarithmic addition and logarithmic subtraction are two fundamental properties of logarithms. Logarithmic addition states that log⁡ax+log⁡ay=log⁡a(xy)\log _a x + \log _a y = \log _a (xy), while logarithmic subtraction states that log⁡ax−log⁡ay=log⁡a(xy)\log _a x - \log _a y = \log _a \left(\frac{x}{y}\right). These properties allow us to combine logarithms with the same base.

Q: How do I apply the logarithmic addition property?

A: To apply the logarithmic addition property, you need to combine the two logarithms on the left-hand side of the equation into a single logarithm. This can be done by multiplying the expressions inside the logarithms.

Q: What is the logarithmic multiplication property?

A: The logarithmic multiplication property states that alog⁡bx=log⁡bxaa \log _b x = \log _b x^a. This property allows us to combine a constant with a logarithm.

Q: How do I apply the logarithmic multiplication property?

A: To apply the logarithmic multiplication property, you need to multiply the constant with the expression inside the logarithm.

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

A: A logarithmic equation is an equation that involves logarithms, while a logarithmic inequality is an inequality that involves logarithms. Logarithmic equations and inequalities are solved using different techniques.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the logarithm on one side of the equation. This can be done by applying the properties of logarithms.

Q: What is the base of a logarithm?

A: The base of a logarithm is the number that is used to define the logarithm. For example, in the logarithm log⁡ax\log _a x, the base is aa.

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

A: To change the base of a logarithm, you need to use the change of base formula: log⁡ax=log⁡bxlog⁡ba\log _a x = \frac{\log _b x}{\log _b a}.

Q: What is the logarithmic identity?

A: The logarithmic identity states that log⁡ax=log⁡bxlog⁡ba\log _a x = \frac{\log _b x}{\log _b a}. This identity allows us to change the base of a logarithm.

Q: How do I apply the logarithmic identity?

A: To apply the logarithmic identity, you need to use the formula log⁡ax=log⁡bxlog⁡ba\log _a x = \frac{\log _b x}{\log _b a}.

Conclusion

In this article, we have answered some frequently asked questions about logarithmic equations. Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. We hope that this article has provided you with a better understanding of logarithmic equations and how to solve them.

References

  • [1] "Logarithmic Properties" by Math Open Reference. Retrieved February 2023.
  • [2] "Logarithmic Equations" by Khan Academy. Retrieved February 2023.
  • [3] "Properties of Logarithms" by Wolfram MathWorld. Retrieved February 2023.

Additional Resources

  • [1] "Logarithmic Equations" by MIT OpenCourseWare. Retrieved February 2023.
  • [2] "Logarithmic Functions" by Mathway. Retrieved February 2023.
  • [3] "Logarithmic Identities" by Purplemath. Retrieved February 2023.