Fill In The Missing Values To Make The Equations True.(a) Log ⁡ 4 3 + Log ⁡ 4 7 = Log ⁡ 4 □ \log_4 3 + \log_4 7 = \log_4 \square Lo G 4 ​ 3 + Lo G 4 ​ 7 = Lo G 4 ​ □ (b) Log ⁡ 9 4 − Log ⁡ 9 □ = Log ⁡ 9 4 5 \log_9 4 - \log_9 \square = \log_9 \frac{4}{5} Lo G 9 ​ 4 − Lo G 9 ​ □ = Lo G 9 ​ 5 4 ​ (c) Log ⁡ 9 9 = □ Log ⁡ 9 3 \log_9 9 = \square \log_9 3 Lo G 9 ​ 9 = □ Lo G 9 ​ 3

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Introduction

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 explore three logarithmic equations and provide step-by-step solutions to each of them. We will also discuss the properties of logarithms and how they can be used to simplify and solve logarithmic equations.

Properties of Logarithms

Before we dive into the solutions, let's review some of the key properties of logarithms:

  • Product Rule: log⁡b(xy)=log⁡bx+log⁡by\log_b (xy) = \log_b x + \log_b y
  • Quotient Rule: log⁡bxy=log⁡bx−log⁡by\log_b \frac{x}{y} = \log_b x - \log_b y
  • Power Rule: log⁡bxy=ylog⁡bx\log_b x^y = y \log_b x
  • Change of Base Formula: log⁡bx=log⁡axlog⁡ab\log_b x = \frac{\log_a x}{\log_a b}

Solution (a)

The first equation is log⁡43+log⁡47=log⁡4□\log_4 3 + \log_4 7 = \log_4 \square. To solve this equation, we can use the product rule, which states that log⁡b(xy)=log⁡bx+log⁡by\log_b (xy) = \log_b x + \log_b y. In this case, we can rewrite the equation as log⁡4(3⋅7)=log⁡4□\log_4 (3 \cdot 7) = \log_4 \square.

Using the product rule, we can simplify the left-hand side of the equation:

log⁡4(3⋅7)=log⁡421\log_4 (3 \cdot 7) = \log_4 21

Now, we can equate the two sides of the equation:

log⁡421=log⁡4□\log_4 21 = \log_4 \square

To solve for □\square, we can use the fact that if log⁡bx=log⁡by\log_b x = \log_b y, then x=yx = y. Therefore, we can conclude that:

□=21\square = 21

Solution (b)

The second equation is log⁡94−log⁡9□=log⁡945\log_9 4 - \log_9 \square = \log_9 \frac{4}{5}. To solve this equation, we can use the quotient rule, which states that log⁡bxy=log⁡bx−log⁡by\log_b \frac{x}{y} = \log_b x - \log_b y. In this case, we can rewrite the equation as log⁡94−log⁡9(□⋅54)=log⁡945\log_9 4 - \log_9 (\square \cdot \frac{5}{4}) = \log_9 \frac{4}{5}.

Using the quotient rule, we can simplify the left-hand side of the equation:

log⁡94−log⁡9(□⋅54)=log⁡945\log_9 4 - \log_9 (\square \cdot \frac{5}{4}) = \log_9 \frac{4}{5}

Now, we can equate the two sides of the equation:

log⁡94−log⁡9(□⋅54)=log⁡945\log_9 4 - \log_9 (\square \cdot \frac{5}{4}) = \log_9 \frac{4}{5}

To solve for □\square, we can use the fact that if log⁡bx=log⁡by\log_b x = \log_b y, then x=yx = y. Therefore, we can conclude that:

□⋅54=45\square \cdot \frac{5}{4} = \frac{4}{5}

Simplifying the equation, we get:

□=45⋅45\square = \frac{4}{5} \cdot \frac{4}{5}

□=1625\square = \frac{16}{25}

Solution (c)

The third equation is log⁡99=□log⁡93\log_9 9 = \square \log_9 3. To solve this equation, we can use the fact that log⁡bb=1\log_b b = 1. Therefore, we can rewrite the equation as:

1=□log⁡931 = \square \log_9 3

Now, we can equate the two sides of the equation:

1=□log⁡931 = \square \log_9 3

To solve for □\square, we can use the fact that if log⁡bx=log⁡by\log_b x = \log_b y, then x=yx = y. Therefore, we can conclude that:

□=1log⁡93\square = \frac{1}{\log_9 3}

Using the change of base formula, we can rewrite the equation as:

□=1log⁡3log⁡9\square = \frac{1}{\frac{\log 3}{\log 9}}

□=log⁡9log⁡3\square = \frac{\log 9}{\log 3}

□=log⁡32log⁡3\square = \frac{\log 3^2}{\log 3}

□=2log⁡3log⁡3\square = \frac{2 \log 3}{\log 3}

□=2\square = 2

Conclusion

In this article, we have solved three logarithmic equations using the properties of logarithms. We have used the product rule, quotient rule, power rule, and change of base formula to simplify and solve each equation. We have also discussed the properties of logarithms and how they can be used to simplify and solve logarithmic equations.

Final Answer

The final answers to the three equations are:

  • log⁡43+log⁡47=log⁡421\log_4 3 + \log_4 7 = \log_4 21
  • log⁡94−log⁡9□=log⁡945\log_9 4 - \log_9 \square = \log_9 \frac{4}{5}, □=1625\square = \frac{16}{25}
  • log⁡99=2log⁡93\log_9 9 = 2 \log_9 3

Introduction

In our previous article, we explored three logarithmic equations and provided step-by-step solutions to each of them. In this article, we will answer some of the most frequently asked questions about logarithmic equations.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves logarithms. Logarithms are the inverse of exponents, and they are used to solve equations that involve exponential expressions.

Q: What are the properties of logarithms?

A: The properties of logarithms are:

  • Product Rule: log⁡b(xy)=log⁡bx+log⁡by\log_b (xy) = \log_b x + \log_b y
  • Quotient Rule: log⁡bxy=log⁡bx−log⁡by\log_b \frac{x}{y} = \log_b x - \log_b y
  • Power Rule: log⁡bxy=ylog⁡bx\log_b x^y = y \log_b x
  • Change of Base Formula: log⁡bx=log⁡axlog⁡ab\log_b x = \frac{\log_a x}{\log_a b}

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. Here are the steps:

  1. Use the product rule to combine logarithms with the same base.
  2. Use the quotient rule to combine logarithms with the same base.
  3. Use the power rule to simplify logarithms with exponents.
  4. Use the change of base formula to change the base of the logarithm.

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

A: A logarithmic equation is an equation that involves logarithms, while an exponential equation is an equation that involves exponents. For example:

  • Logarithmic equation: log⁡2x=3\log_2 x = 3
  • Exponential equation: 23=x2^3 = x

Q: Can I use a calculator to solve logarithmic equations?

A: Yes, you can use a calculator to solve logarithmic equations. However, it's always a good idea to check your work by plugging the solution back into the original equation.

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

A: Here are some common mistakes to avoid when solving logarithmic equations:

  • Forgetting to use the product rule or quotient rule when combining logarithms.
  • Forgetting to use the power rule when simplifying logarithms with exponents.
  • Forgetting to change the base of the logarithm when using the change of base formula.
  • Not checking your work by plugging the solution back into the original equation.

Q: Can I use logarithmic equations to solve real-world problems?

A: Yes, logarithmic equations can be used to solve real-world problems. For example:

  • Modeling population growth: log⁡2P=t\log_2 P = t
  • Modeling chemical reactions: log⁡10[A]=−kt\log_10 [A] = -kt
  • Modeling financial growth: log⁡2A=rt\log_2 A = rt

Conclusion

In this article, we have answered some of the most frequently asked questions about logarithmic equations. We hope that this article has provided a clear and concise explanation of how to solve logarithmic equations and has helped to clarify any confusion. If you have any further questions or need further clarification, please don't hesitate to ask.

Final Tips

  • Always use the product rule and quotient rule when combining logarithms.
  • Always use the power rule when simplifying logarithms with exponents.
  • Always change the base of the logarithm when using the change of base formula.
  • Always check your work by plugging the solution back into the original equation.

By following these tips and practicing regularly, you will become proficient in solving logarithmic equations and be able to apply them to real-world problems.