Solve The Equation $\log_6(13-x) = 1$.$x = \square$

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Introduction

In this article, we will delve into solving a logarithmic equation, specifically the equation $\log_6(13-x) = 1$. This type of equation involves logarithms, which are a fundamental concept in mathematics. Logarithms are used to solve equations that involve exponential functions, and they have numerous applications in various fields, including science, engineering, and finance.

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithm. The logarithm of a number is the exponent to which a base number must be raised to produce that number. For example, $\log_6(36) = 2$ because $6^2 = 36$. In the equation $\log_6(13-x) = 1$, we are looking for the value of $x$ that makes the logarithm equal to 1.

Solving the Equation

To solve the equation $\log_6(13-x) = 1$, we need to get rid of the logarithm. We can do this by using the definition of a logarithm. Since $\log_6(13-x) = 1$, we know that $6^1 = 13-x$. Therefore, we can rewrite the equation as:

$$6^1 = 13-x$$

Simplifying the Equation

Now, we can simplify the equation by evaluating the left-hand side. Since $6^1 = 6$, we can rewrite the equation as:

$$6 = 13-x$$

Isolating the Variable

To isolate the variable $x$, we need to get rid of the constant term on the right-hand side. We can do this by subtracting 13 from both sides of the equation:

$$-7 = -x$$

Solving for $x$

Finally, we can solve for $x$ by multiplying both sides of the equation by -1:

$$x = 7$$

Conclusion

In this article, we solved the equation $\log_6(13-x) = 1$ by using the definition of a logarithm and simplifying the equation. We found that the value of $x$ that satisfies the equation is $x = 7$. This type of problem is an example of a logarithmic equation, and it requires a good understanding of logarithmic functions and their properties.

Applications of Logarithmic Equations

Logarithmic equations have numerous applications in various fields, including science, engineering, and finance. For example, logarithmic equations are used to model population growth, chemical reactions, and financial transactions. They are also used to solve problems involving exponential decay and growth.

Real-World Examples

Here are a few real-world examples of logarithmic equations:

• A company's stock price is increasing exponentially, and the logarithmic equation $\log_2(1000) = t$ represents the time it takes for the stock price to reach $1000.
• A population of bacteria is growing exponentially, and the logarithmic equation $\log_3(1000) = t$ represents the time it takes for the population to reach $1000$.
• A financial transaction involves an exponential decay, and the logarithmic equation $\log_4(1000) = t$ represents the time it takes for the transaction to complete.

Tips and Tricks

Here are a few tips and tricks for solving logarithmic equations:

• Use the definition of a logarithm to rewrite the equation.
• Simplify the equation by evaluating the left-hand side.
• Isolate the variable by subtracting or adding constants.
• Solve for the variable by multiplying or dividing both sides of the equation.

Common Mistakes

Here are a few common mistakes to avoid when solving logarithmic equations:

• Not using the definition of a logarithm to rewrite the equation.
• Not simplifying the equation by evaluating the left-hand side.
• Not isolating the variable by subtracting or adding constants.
• Not solving for the variable by multiplying or dividing both sides of the equation.

Conclusion

In conclusion, solving logarithmic equations requires a good understanding of logarithmic functions and their properties. By using the definition of a logarithm and simplifying the equation, we can solve for the variable and find the solution to the equation. Logarithmic equations have numerous applications in various fields, and they are used to model population growth, chemical reactions, and financial transactions. By following the tips and tricks outlined in this article, we can avoid common mistakes and solve logarithmic equations with ease.

Introduction

In our previous article, we solved the equation $\log_6(13-x) = 1$ and found that the value of $x$ that satisfies the equation is $x = 7$. In this article, we will answer some frequently asked questions about logarithmic equations.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm. The logarithm of a number is the exponent to which a base number must be raised to produce that number.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to get rid of the logarithm. You can do this by using the definition of a logarithm and simplifying the equation.

Q: What is the definition of a logarithm?

A: The definition of a logarithm is that $\log_b(a) = c$ is equivalent to $b^c = a$. For example, $\log_6(36) = 2$ because $6^2 = 36$.

Q: How do I simplify a logarithmic equation?

A: To simplify a logarithmic equation, you need to evaluate the left-hand side of the equation. For example, if you have the equation $\log_6(36) = 2$, you can simplify it by evaluating the left-hand side: $6^2 = 36$.

Q: How do I isolate the variable in a logarithmic equation?

A: To isolate the variable in a logarithmic equation, you need to get rid of the constant term on the right-hand side of the equation. You can do this by subtracting or adding constants.

Q: How do I solve for the variable in a logarithmic equation?

A: To solve for the variable in a logarithmic equation, you need to multiply or divide both sides of the equation by a constant.

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

A: Some common mistakes to avoid when solving logarithmic equations include not using the definition of a logarithm to rewrite the equation, not simplifying the equation by evaluating the left-hand side, not isolating the variable by subtracting or adding constants, and not solving for the variable by multiplying or dividing both sides of the equation.

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

A: Logarithmic equations have numerous applications in various fields, including science, engineering, and finance. For example, logarithmic equations are used to model population growth, chemical reactions, and financial transactions.

Q: How do I use logarithmic equations to model population growth?

A: To use logarithmic equations to model population growth, you need to use the equation $P(t) = P_0 e^{kt}$, where $P(t)$ is the population at time $t$, $P_0$ is the initial population, $k$ is the growth rate, and $t$ is time.

Q: How do I use logarithmic equations to model chemical reactions?

A: To use logarithmic equations to model chemical reactions, you need to use the equation $A(t) = A_0 e^{kt}$, where $A(t)$ is the amount of substance $A$ at time $t$, $A_0$ is the initial amount of substance $A$, $k$ is the reaction rate, and $t$ is time.

Q: How do I use logarithmic equations to model financial transactions?

A: To use logarithmic equations to model financial transactions, you need to use the equation $V(t) = V_0 e^{kt}$, where $V(t)$ is the value of the transaction at time $t$, $V_0$ is the initial value of the transaction, $k$ is the interest rate, and $t$ is time.

Conclusion

In conclusion, logarithmic equations are a powerful tool for modeling real-world phenomena. By understanding the definition of a logarithm and how to simplify and solve logarithmic equations, you can use logarithmic equations to model population growth, chemical reactions, and financial transactions.