Solve For $x$:$\log (x+5) - \log (x+2) = 1$$ X = X = X = [/tex] $\square$You May Enter The Exact Value Or Round To 4 Decimal Places.

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Introduction

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Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving a specific logarithmic equation involving the subtraction of two logarithms. The equation is given as:

$$\log (x+5) - \log (x+2) = 1$$

Our goal is to find the value of $x$ that satisfies this equation.

Understanding Logarithmic Properties

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Before we dive into solving the equation, it's essential to understand the properties of logarithms. One of the most important properties is the subtraction rule, which states that:

$$\log_a (b) - \log_a (c) = \log_a \left(\frac{b}{c}\right)$$

This property allows us to combine the two logarithms in the given equation into a single logarithm.

Applying the Subtraction Rule

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Using the subtraction rule, we can rewrite the equation as:

$$\log \left(\frac{x+5}{x+2}\right) = 1$$

This simplifies the equation and makes it easier to solve.

Exponentiating Both Sides

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To get rid of the logarithm, we can exponentiate both sides of the equation. Since the base of the logarithm is not specified, we will assume it is the natural logarithm (base $e$). Exponentiating both sides gives us:

$$e^{\log \left(\frac{x+5}{x+2}\right)} = e^1$$

Using the property that $e^{\log x} = x$, we can simplify the left-hand side of the equation:

$$\frac{x+5}{x+2} = e$$

Solving for $x$

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Now we have a simple algebraic equation to solve for $x$. We can start by multiplying both sides of the equation by $x+2$ to get rid of the fraction:

$$(x+5) = e(x+2)$$

Expanding the right-hand side gives us:

$$x+5 = ex + 2e$$

Subtracting $x$ from both sides gives us:

$$5 = ex - 2e$$

Adding $2e$ to both sides gives us:

$$5 + 2e = ex$$

Dividing both sides by $e$ gives us:

$$\frac{5 + 2e}{e} = x$$

Simplifying the Expression

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To simplify the expression, we can use the fact that $e$ is a constant. We can rewrite the expression as:

$$x = \frac{5 + 2e}{e}$$

Using a calculator, we can find the value of $x$:

$$x \approx \frac{5 + 2(2.718)}{2.718}$$

$$x \approx \frac{5 + 5.436}{2.718}$$

$$x \approx \frac{10.436}{2.718}$$

$$x \approx 3.82$$

Conclusion

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In this article, we solved a logarithmic equation involving the subtraction of two logarithms. We used the subtraction rule to combine the two logarithms into a single logarithm, and then exponentiated both sides to get rid of the logarithm. We then solved for $x$ using algebraic manipulations. The final value of $x$ is approximately $3.82$.

Final Answer

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The final answer is: $\boxed{3.82}$

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Introduction

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In our previous article, we solved a logarithmic equation involving the subtraction of two logarithms. We received many questions from readers who were struggling to understand the concept of logarithmic equations. In this article, we will address some of the most frequently asked questions about solving logarithmic equations.

Q&A

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

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. An exponential equation, on the other hand, is an equation that involves an exponent. For example, the equation $\log (x+5) - \log (x+2) = 1$ is a logarithmic equation, while the equation $2^x = 8$ is an exponential equation.

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

A: When solving a logarithmic equation, you can use any base that is convenient for you. However, it's often easiest to use the natural logarithm (base $e$) or the common logarithm (base 10). If you're not sure which base to use, you can try using the natural logarithm first and then convert to the desired base later.

Q: Can I use the same properties of logarithms to solve exponential equations?

A: No, the properties of logarithms are not the same as the properties of exponents. While both logarithms and exponents involve powers of a base, the rules for manipulating them are different. For example, the property $\log_a (b) - \log_a (c) = \log_a \left(\frac{b}{c}\right)$ does not apply to exponential equations.

Q: How do I deal with logarithmic equations that involve fractions or decimals?

A: When dealing with logarithmic equations that involve fractions or decimals, it's often easiest to get rid of the fraction or decimal by multiplying both sides of the equation by a power of the base. For example, if you have the equation $\log \left(\frac{x+5}{x+2}\right) = 1$, you can multiply both sides by $x+2$ to get rid of the fraction.

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

A: Yes, you can use a calculator to solve logarithmic equations. However, keep in mind that calculators may not always give you the exact answer, especially if the equation involves complex numbers or irrational numbers. It's always a good idea to check your answer by plugging it back into the original equation.

Q: How do I know if my answer is correct?

A: To check if your answer is correct, plug it back into the original equation and simplify. If the equation holds true, then your answer is correct. If not, then you need to go back and recheck your work.

Common Mistakes to Avoid

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When solving logarithmic equations, there are several common mistakes to avoid. Here are a few:

• Not using the correct properties of logarithms: Make sure you're using the correct properties of logarithms, such as the subtraction rule and the power rule.
• Not getting rid of fractions or decimals: Make sure you're getting rid of fractions or decimals by multiplying both sides of the equation by a power of the base.
• Not checking your answer: Make sure you're plugging your answer back into the original equation to check if it's correct.

Conclusion

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Solving logarithmic equations can be challenging, but with the right approach and a little practice, you can master them. Remember to use the correct properties of logarithms, get rid of fractions or decimals, and check your answer. With these tips and a little patience, you'll be solving logarithmic equations like a pro in no time.

Final Tips

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Here are a few final tips to keep in mind when solving logarithmic equations:

• Practice, practice, practice: The more you practice solving logarithmic equations, the more comfortable you'll become with the concepts and techniques.
• Use online resources: There are many online resources available that can help you practice solving logarithmic equations, including video tutorials and practice problems.
• Seek help when needed: Don't be afraid to ask for help if you're struggling with a particular problem or concept.