Solve The Equation Log ⁡ 6 ( 13 − X ) = 1 \log_6(13-x)=1 Lo G 6 ( 13 − X ) = 1 . X = X = X =

Mar 11, 2025 by ADMIN 95 views

$Solve the Equation $\log_6(13-x)=1$$

Introduction

In this article, we will delve into the world of logarithms and solve a logarithmic equation. The equation we will be solving is $\log_6(13-x)=1$ . This equation involves a logarithm with base 6, and our goal is to find the value of $x$ that satisfies this equation. We will use the properties of logarithms and algebraic manipulations to solve for $x$ .

Understanding Logarithms

Before we dive into solving the equation, let's take a moment to understand what logarithms are. A logarithm is the inverse operation of exponentiation. In other words, if we have a number $y$ that is the result of raising a base $b$ to a power $x$ , then the logarithm of $y$ with base $b$ is equal to $x$ . This can be expressed mathematically as:

b^x = y \Rightarrow \log_b(y) = x

For example, if we have $2^3 = 8$ , then the logarithm of 8 with base 2 is equal to 3, since $\log_2(8) = 3$ .

Solving the Equation

Now that we have a basic understanding of logarithms, let's turn our attention to solving the equation $\log_6(13-x)=1$ . To solve this equation, we can use the definition of logarithms to rewrite the equation in exponential form.

\log_6(13-x) = 1 \Rightarrow 6^1 = 13-x

Simplifying the right-hand side of the equation, we get:

6 = 13-x

Now, we can solve for $x$ by isolating it on one side of the equation. Subtracting 13 from both sides of the equation, we get:

-7 = -x

Multiplying both sides of the equation by -1, we get:

7 = x

Therefore, the value of $x$ that satisfies the equation $\log_6(13-x)=1$ is $x = 7$ .

Conclusion

In this article, we solved the logarithmic equation $\log_6(13-x)=1$ using the properties of logarithms and algebraic manipulations. We first rewrote the equation in exponential form, and then solved for $x$ by isolating it on one side of the equation. The value of $x$ that satisfies the equation is $x = 7$ . This problem demonstrates the importance of understanding logarithms and how to apply them to solve equations.

Additional Examples

Here are a few more examples of logarithmic equations that can be solved using the same techniques:

$\log_4(9-x) = 2$
$\log_3(12-x) = 1$
$\log_2(15-x) = 3$

These equations can be solved using the same steps as the original equation, and the solutions can be found by isolating $x$ on one side of the equation.

Tips and Tricks

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

Always start by rewriting the equation in exponential form.
Use the properties of logarithms to simplify the equation.
Isolate $x$ on one side of the equation by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
Check your solution by plugging it back into the original equation.

By following these tips and tricks, you can solve logarithmic equations with ease and confidence.

Common Mistakes

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

Not rewriting the equation in exponential form.
Not using the properties of logarithms to simplify the equation.
Not isolating $x$ on one side of the equation.
Not checking the solution by plugging it back into the original equation.

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

Real-World Applications

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.

By understanding logarithmic equations and how to solve them, you can apply this knowledge to real-world problems and make informed decisions.

Conclusion

In conclusion, solving logarithmic equations is an important skill that can be applied to a wide range of problems. By understanding the properties of logarithms and using algebraic manipulations, you can solve logarithmic equations with ease and confidence. Whether you are working in finance, science, or engineering, logarithmic equations are an essential tool for solving problems and making informed decisions.

Introduction

In our previous article, we solved the logarithmic equation $\log_6(13-x)=1$ using the properties of logarithms and algebraic manipulations. In this article, we will answer some frequently asked questions about logarithmic equations and provide additional examples and tips for solving them.

Q&A

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, $\log_6(13-x)=1$ is a logarithmic equation, while $6^x = 13-x$ is an exponential equation.

Q: How do I know which base to use when solving a logarithmic equation?

A: The base of a logarithmic equation is usually given in the problem statement. If the base is not given, you can choose any base that is convenient for you. However, it's usually best to choose a base that is a power of 10, such as 2, 5, or 10.

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

A: Yes, you can use a calculator to solve a logarithmic equation. However, it's usually best to use a calculator to check your solution, rather than relying on it to solve the equation.

Q: How do I know if a logarithmic equation has a solution?

A: A logarithmic equation has a solution if and only if the argument of the logarithm is positive. For example, the equation $\log_6(13-x)=1$ has a solution if and only if $13-x > 0$ , which is equivalent to $x < 13$ .

Q: Can I use logarithmic equations to solve exponential equations?

A: Yes, you can use logarithmic equations to solve exponential equations. For example, if you have the equation $6^x = 13-x$ , you can take the logarithm of both sides to get $\log_6(6^x) = \log_6(13-x)$ , which simplifies to $x = \log_6(13-x)$ .

Q: How do I know if a logarithmic equation is true or false?

A: A logarithmic equation is true if and only if the argument of the logarithm is equal to the base raised to the power of the logarithm. For example, the equation $\log_6(13-x)=1$ is true if and only if $13-x = 6^1$ , which is equivalent to $13-x = 6$ .