Solve For $x$. Log ⁡ 5 ( − 2 X + 5 ) = 2 \log _5(-2x + 5) = 2 Lo G 5 ​ ( − 2 X + 5 ) = 2

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Understanding the Problem

Logarithmic equations can be challenging to solve, especially when they involve variables within the logarithmic function. In this problem, we are given the equation $\log_5(-2x + 5) = 2$, and we need to solve for the variable $x$. To start, let's recall the definition of a logarithm: if $y = \log_b(x)$, then $b^y = x$. In this case, we have $\log_5(-2x + 5) = 2$, which means that $5^2 = -2x + 5$.

Solving the Equation

To solve for $x$, we need to isolate the variable on one side of the equation. Let's start by evaluating the expression $5^2$, which is equal to $25$. So, we have:

$$25 = -2x + 5$$

Next, we can subtract $5$ from both sides of the equation to get:

$$20 = -2x$$

Isolating the Variable

Now, we need to isolate the variable $x$ on one side of the equation. To do this, we can divide both sides of the equation by $-2$, which gives us:

$$x = -10$$

Checking the Solution

Before we conclude that $x = -10$ is the solution to the equation, let's check our work by plugging the value of $x$ back into the original equation. If $x = -10$, then we have:

$$\log_5(-2(-10) + 5) = \log_5(20 + 5) = \log_5(25)$$

Since $\log_5(25) = 2$, we can see that our solution is correct.

Conclusion

In this problem, we used the definition of a logarithm to solve the equation $\log_5(-2x + 5) = 2$. We started by evaluating the expression $5^2$, which is equal to $25$. We then subtracted $5$ from both sides of the equation to get $20 = -2x$, and finally, we divided both sides of the equation by $-2$ to get $x = -10$. We checked our work by plugging the value of $x$ back into the original equation, and we found that our solution is correct.

Tips and Tricks

• When solving logarithmic equations, it's essential to remember the definition of a logarithm: if $y = \log_b(x)$, then $b^y = x$.
• To solve for $x$, you need to isolate the variable on one side of the equation.
• When checking your work, make sure to plug the value of $x$ back into the original equation to ensure that it's true.

Common Mistakes

• Failing to evaluate the expression inside the logarithmic function.
• Not isolating the variable on one side of the equation.
• Not checking the solution by plugging the value of $x$ back into the original equation.

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.

Practice Problems

Try solving the following logarithmic equations:

• $\log_3(x + 2) = 1$
• $\log_2(x - 1) = 3$
• $\log_4(x + 1) = 2$

Conclusion

Solving logarithmic equations requires a deep understanding of the definition of a logarithm and the properties of logarithmic functions. By following the steps outlined in this article, you can solve even the most challenging logarithmic equations. Remember to evaluate the expression inside the logarithmic function, isolate the variable on one side of the equation, and check your work by plugging the value of $x$ back into the original equation. With practice and patience, you'll become a master of solving logarithmic equations.

Frequently Asked Questions

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithmic function, which is a function that takes a number as input and returns the power 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 isolate the variable on one side of the equation. This can be done by using the properties of logarithms, such as the product rule and the quotient rule.

Q: What is the product rule for logarithms?

A: The product rule for logarithms states that $\log_b(xy) = \log_b(x) + \log_b(y)$.

Q: What is the quotient rule for logarithms?

A: The quotient rule for logarithms states that $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$.

Q: How do I use the product rule and quotient rule to solve a logarithmic equation?

A: To use the product rule and quotient rule to solve a logarithmic equation, you need to apply the rules to the equation and then isolate the variable on one side of the equation.

Q: What is the definition of a logarithm?

A: The definition of a logarithm is: if $y = \log_b(x)$, then $b^y = x$.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to use the definition of a logarithm and the properties of logarithms.

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

A: Some common mistakes to avoid when solving logarithmic equations include:

• Failing to evaluate the expression inside the logarithmic function.
• Not isolating the variable on one side of the equation.
• Not checking the solution by plugging the value of $x$ back into the original equation.

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

A: To check your work when solving a logarithmic equation, you need to plug the value of $x$ back into the original equation and verify that it is true.

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

A: Some real-world applications of logarithmic equations include:

• 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 practice solving logarithmic equations?

A: To practice solving logarithmic equations, you can try solving the following problems:

• $\log_3(x + 2) = 1$
• $\log_2(x - 1) = 3$
• $\log_4(x + 1) = 2$

Q: What are some tips for solving logarithmic equations?

A: Some tips for solving logarithmic equations include:

• Start by evaluating the expression inside the logarithmic function.
• Use the properties of logarithms to simplify the equation.
• Isolate the variable on one side of the equation.
• Check your work by plugging the value of $x$ back into the original equation.

Conclusion

Solving logarithmic equations requires a deep understanding of the definition of a logarithm and the properties of logarithmic functions. By following the steps outlined in this article, you can solve even the most challenging logarithmic equations. Remember to evaluate the expression inside the logarithmic function, use the properties of logarithms to simplify the equation, isolate the variable on one side of the equation, and check your work by plugging the value of $x$ back into the original equation. With practice and patience, you'll become a master of solving logarithmic equations.