Fill In The Missing Values To Make The Equations True.(a) Log ⁡ 7 3 + Log ⁡ 7 8 = Log ⁡ 7 \log_7 3 + \log_7 8 = \log_7 Lo G 7 ​ 3 + Lo G 7 ​ 8 = Lo G 7 ​ □ \square □ (b) Log ⁡ 3 7 − Log ⁡ 3 \log_3 7 - \log_3 Lo G 3 ​ 7 − Lo G 3 ​ □ = Log ⁡ 3 7 5 \square = \log_3 \frac{7}{5} □ = Lo G 3 ​ 5 7 ​ (c) Log ⁡ 9 1 9 = − 2 Log ⁡ 9 \log_9 \frac{1}{9} = -2 \log_9 Lo G 9 ​ 9 1 ​ = − 2 Lo G 9 ​ □ \square □

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.

Equation (a): $\log_7 3 + \log_7 8 = \log_7$ $\square$

To solve this equation, we need to use the property of logarithms that states $\log_a b + \log_a c = \log_a (b \cdot c)$. This property allows us to combine the two logarithmic terms on the left-hand side of the equation into a single logarithmic term.

$\log_7 3 + \log_7 8 = \log_7 (3 \cdot 8)$

Using the property of logarithms that states $\log_a b = c \implies b = a^c$, we can rewrite the equation as:

$\log_7 (3 \cdot 8) = \log_7 x$

where $x$ is the value we are trying to find.

To solve for $x$, we can use the fact that if $\log_a b = \log_a c$, then $b = c$. Therefore, we can equate the arguments of the logarithmic terms:

$3 \cdot 8 = x$

Simplifying the equation, we get:

$24 = x$

Therefore, the value of $x$ is 24.

Equation (b): $\log_3 7 - \log_3$ $\square = \log_3 \frac{7}{5}$

To solve this equation, we need to use the property of logarithms that states $\log_a b - \log_a c = \log_a \left(\frac{b}{c}\right)$. This property allows us to combine the two logarithmic terms on the left-hand side of the equation into a single logarithmic term.

$\log_3 7 - \log_3 x = \log_3 \frac{7}{5}$

Using the property of logarithms that states $\log_a b = c \implies b = a^c$, we can rewrite the equation as:

$\log_3 \left(\frac{7}{x}\right) = \log_3 \frac{7}{5}$

To solve for $x$, we can use the fact that if $\log_a b = \log_a c$, then $b = c$. Therefore, we can equate the arguments of the logarithmic terms:

$\frac{7}{x} = \frac{7}{5}$

Simplifying the equation, we get:

$x = 5$

Therefore, the value of $x$ is 5.

Equation (c): $\log_9 \frac{1}{9} = -2 \log_9$ $\square$

To solve this equation, we need to use the property of logarithms that states $\log_a b = c \implies b = a^c$. This property allows us to rewrite the equation as:

$\log_9 \frac{1}{9} = -2 \log_9 y$

where $y$ is the value we are trying to find.

Using the fact that $\log_a b = c \implies b = a^c$, we can rewrite the equation as:

$\frac{1}{9} = 9^{-2} \cdot y$

Simplifying the equation, we get:

$\frac{1}{9} = \frac{y}{81}$

To solve for $y$, we can multiply both sides of the equation by 81:

$y = 81 \cdot \frac{1}{9}$

Simplifying the equation, we get:

$y = 9$

Therefore, the value of $y$ is 9.

Conclusion

In this article, we have explored three logarithmic equations and provided step-by-step solutions to each of them. We have also discussed the properties of logarithms and how they can be used to simplify and solve logarithmic equations. By understanding the properties of logarithms and how to apply them, we can solve a wide range of logarithmic equations and gain a deeper understanding of the underlying mathematics.

Properties of Logarithms

Logarithmic equations are based on the properties of logarithms, which are as follows:

• Product Rule: $\log_a b + \log_a c = \log_a (b \cdot c)$
• Quotient Rule: $\log_a b - \log_a c = \log_a \left(\frac{b}{c}\right)$
• Power Rule: $\log_a b^c = c \log_a b$
• Change of Base Rule: $\log_a b = \frac{\log_c b}{\log_c a}$

These properties are essential for solving logarithmic equations and can be used to simplify and solve a wide range of logarithmic equations.

Real-World Applications

Logarithmic equations have a wide range of 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.

By understanding the properties of logarithms and how to apply them, we can solve a wide range of logarithmic equations and gain a deeper understanding of the underlying mathematics.

Final Thoughts

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 are used to solve problems that involve exponential growth or decay.

Q: What are the properties of logarithms?

A: The properties of logarithms are:

• Product Rule: $\log_a b + \log_a c = \log_a (b \cdot c)$
• Quotient Rule: $\log_a b - \log_a c = \log_a \left(\frac{b}{c}\right)$
• Power Rule: $\log_a b^c = c \log_a b$
• 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, you need to use the properties of logarithms to simplify the equation and isolate the variable. Here are the steps:

1. Use the product rule or quotient rule to combine the logarithmic terms.
2. Use the power rule to simplify the equation.
3. Use the change of base rule to change the base of the logarithm.
4. Solve for the variable.

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

A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent. For example:

• Logarithmic equation: $\log_2 x = 3$
• Exponential equation: $2^x = 8$

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you need to use a graphing calculator or a computer program. Here are the steps:

1. Enter the function into the graphing calculator or computer program.
2. Set the window to the correct range.
3. Graph the function.
4. Analyze the graph to determine the behavior of the function.

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.

Q: How do I use logarithmic equations in real-world problems?

A: To use logarithmic equations in real-world problems, you need to:

1. Identify the problem and determine the type of logarithmic equation needed.
2. Use the properties of logarithms to simplify the equation.
3. Solve for the variable.
4. Interpret the results in the context of the problem.

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 product rule or quotient rule: Make sure to use the product rule or quotient rule to combine the logarithmic terms.
• Forgetting to use the power rule: Make sure to use the power rule to simplify the equation.
• Forgetting to use the change of base rule: Make sure to use the change of base rule to change the base of the logarithm.
• Not checking the domain: Make sure to check the domain of the logarithmic function to ensure that it is defined.

Q: How do I check my work when solving logarithmic equations?

A: To check your work when solving logarithmic equations, you need to:

1. Plug in the solution into the original equation.
2. Simplify the equation.
3. Check that the equation is true.
4. Interpret the results in the context of the problem.

By following these steps and avoiding common mistakes, you can ensure that your work is accurate and complete.