Solving Logarithmic Equations Step By Step With Example

Hey guys! 👋 Ever found yourself scratching your head over a logarithmic equation? Don't worry; you're not alone! Logarithmic equations can seem tricky at first, but with a bit of know-how, you can totally nail them. In this guide, we're going to break down the steps to solve a specific logarithmic equation and equip you with the knowledge to tackle similar problems. So, let's dive in and make those logs feel less like a puzzle and more like a piece of cake! 🍰

Understanding Logarithmic Equations

Before we jump into solving the equation, let's quickly recap what logarithmic equations are all about. Logarithmic equations are equations where the logarithm of an expression appears. Remember, logarithms are essentially the inverse of exponentiation. So, if we have an equation like $\log_b(x) = y$, it means that $b^y = x$. Understanding this relationship is key to solving logarithmic equations. When dealing with logarithmic equations, our main goal is to isolate the logarithmic terms, combine them if possible, and then convert the equation into its exponential form. This allows us to eliminate the logarithm and solve for the variable. It’s also super important to check our solutions at the end because logarithms have domain restrictions – we can only take the logarithm of positive numbers. This means we need to make sure our solutions don’t lead to taking the logarithm of a negative number or zero, which would make the original equation undefined.

Key Properties and Rules

To effectively solve logarithmic equations, it's essential to have a grasp of some fundamental logarithmic properties. These properties act as tools in our problem-solving toolkit, allowing us to manipulate and simplify equations. Let’s go through some of the most useful ones:

1. Product Rule: The logarithm of the product of two numbers is equal to the sum of their logarithms. Mathematically, this is expressed as $\log_b(MN) = \log_b(M) + \log_b(N)$. This rule is super handy when you have logarithms of products and want to break them down into simpler terms.

2. Quotient Rule: The logarithm of the quotient of two numbers is equal to the difference of their logarithms. Formally, $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$. This is the go-to rule when you're dealing with logarithms of fractions.

3. Power Rule: The logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. In equation form, $\log_b(M^p) = p \log_b(M)$. This rule is incredibly useful for simplifying logarithms with exponents.

4. Change of Base Rule: This rule allows us to convert logarithms from one base to another, which is particularly useful when using calculators that may only have common logarithms (base 10) or natural logarithms (base e). The rule states $\log_b(M) = \frac{\log_c(M)}{\log_c(b)}$, where c is the new base.

5. Logarithm of the Base: The logarithm of the base to itself is always 1. That is, $\log_b(b) = 1$. This is a straightforward but crucial property.

6. Logarithm of 1: The logarithm of 1 in any base is always 0. In other words, $\log_b(1) = 0$. This one pops up more often than you might think!

7. Inverse Properties: These properties highlight the inverse relationship between logarithms and exponentiation:

   • $b^{\log_b(x)} = x$
   • $\log_b(b^x) = x$

Understanding and being able to apply these properties will make solving logarithmic equations much smoother. They help us to simplify complex expressions, combine or separate logarithmic terms, and ultimately, isolate the variable we're solving for. So, keep these rules in your back pocket – you'll be reaching for them often!

Solving the Equation: $\log (2 x+1)=1+\log (x-3)$

Alright, let's get our hands dirty and solve this logarithmic equation step by step. Our mission is to find the value(s) of x that make this equation true. Remember, the key is to isolate the logarithmic terms, combine them if we can, and then transform the equation into exponential form. Buckle up, let’s do this!

Step 1: Rearrange the Equation

First up, we want to gather all the logarithmic terms on one side of the equation. This makes our job of combining them much easier. We have:

$\log (2 x+1)=1+\log (x-3)$

Let’s subtract $\log (x-3)$ from both sides:

$\log (2 x+1) - \log (x-3) = 1$

Now we have all our logarithmic terms happily sitting on the left side, and the constant term on the right. Time to move on to the next step!

Step 2: Combine Logarithmic Terms

Now that we have the logarithms on one side, we can use the quotient rule to combine them. Remember, the quotient rule states that $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$. Applying this rule to our equation, we get:

$\log(\frac{2x+1}{x-3}) = 1$

This looks much cleaner, doesn’t it? We’ve transformed two logarithmic terms into a single one, which is exactly what we wanted. Note that we're dealing with common logarithms here, which means the base is 10.

Step 3: Convert to Exponential Form

Here comes the fun part – converting our logarithmic equation into its exponential form. Recall that the logarithmic equation $\log_b(x) = y$ is equivalent to the exponential equation $b^y = x$. In our case, we have $\log_{10}(\frac{2x+1}{x-3}) = 1$. So, converting this to exponential form, we get:

$10^1 = \frac{2x+1}{x-3}$

This simplifies to:

$10 = \frac{2x+1}{x-3}$

We’ve successfully eliminated the logarithm! Now we’re dealing with a good ol' algebraic equation, which we can solve using familiar methods.

Step 4: Solve for x

To solve for x, we need to get rid of the fraction. We can do this by multiplying both sides of the equation by $(x-3)$:

$10(x-3) = 2x + 1$

Now, let’s distribute the 10 on the left side:

$10x - 30 = 2x + 1$

Next, we want to get all the x terms on one side and the constants on the other. Subtract $2x$ from both sides:

$8x - 30 = 1$

Add 30 to both sides:

$8x = 31$

Finally, divide both sides by 8:

$x = \frac{31}{8}$

Woohoo! We’ve found a potential solution for x. But hold your horses – we're not done yet. Remember, we need to check our solution to make sure it doesn’t lead to taking the logarithm of a negative number or zero.

Step 5: Check the Solution

This is a crucial step that can’t be skipped. Logarithmic functions have a domain restriction: the argument of the logarithm must be positive. We need to plug our solution, $x = \frac{31}{8}$, back into the original equation and make sure that the expressions inside the logarithms are positive.

Let’s check $2x + 1$:

$2(\frac{31}{8}) + 1 = \frac{31}{4} + 1 = \frac{31}{4} + \frac{4}{4} = \frac{35}{4}$

This is positive, so far so good.

Now, let’s check $x - 3$:

$\frac{31}{8} - 3 = \frac{31}{8} - \frac{24}{8} = \frac{7}{8}$

This is also positive! Since both expressions inside the logarithms are positive when $x = \frac{31}{8}$, this solution is valid.

Step 6: State the Solution Set

We’ve done the hard work, and now we can confidently state our solution. The solution set for the equation $\log (2 x+1)=1+\log (x-3)$ is:

${\{\frac{31}{8}\}}$.

We did it! 🎉 Solving logarithmic equations might seem daunting at first, but breaking it down into steps makes it totally manageable. Remember to rearrange, combine logarithms, convert to exponential form, solve for the variable, and, most importantly, check your solution. Keep practicing, and you'll become a log equation-solving pro in no time!

Common Mistakes to Avoid

When solving logarithmic equations, it's easy to stumble upon a few common pitfalls. Being aware of these mistakes can save you a lot of headaches and ensure you arrive at the correct solution. Let's highlight some of the most frequent errors:

1. Forgetting to Check for Extraneous Solutions: This is by far the most common mistake. Logarithmic functions have domain restrictions, meaning the argument (the expression inside the logarithm) must be positive. When solving, you might find solutions that seem correct algebraically but lead to taking the logarithm of a negative number or zero in the original equation. These are called extraneous solutions, and they must be discarded. Always plug your solutions back into the original equation to check if they're valid.

2. Incorrectly Applying Logarithmic Properties: The logarithmic properties (product rule, quotient rule, power rule) are powerful tools, but they must be applied correctly. A common mistake is to misapply these rules, for instance, assuming $\log(A + B)$ is equal to $\log(A) + \log(B)$, which is incorrect. Remember, the product rule applies to the logarithm of a product, not the logarithm of a sum. Review the properties carefully and practice applying them correctly.

3. Dropping the Logarithm Too Early: Sometimes, students try to get rid of the logarithm before they've properly isolated and combined the logarithmic terms. You should only convert the equation to exponential form once you have a single logarithmic expression on one side of the equation. Prematurely dropping the logarithm can lead to an incorrect equation and, consequently, a wrong solution.

4. Ignoring the Base of the Logarithm: When converting from logarithmic to exponential form, it's crucial to remember the base of the logarithm. If no base is written, it's assumed to be 10 (common logarithm). If you forget the base, you'll end up with the wrong exponential equation. For instance, $\log(x) = 2$ means $\log_{10}(x) = 2$, which converts to $10^2 = x$, not $e^2 = x$ (which would be the case for the natural logarithm).

5. Arithmetic Errors: As with any algebraic problem, arithmetic errors can derail your solution. Be careful when performing operations such as distributing, combining like terms, and solving for x. A small mistake in arithmetic can lead to a completely wrong answer.

6. Not Simplifying the Equation: Before diving into solving, take a moment to simplify the equation as much as possible. This might involve combining like terms, distributing, or using algebraic identities. A simplified equation is easier to work with and reduces the chances of making errors.

7. Misunderstanding the Definition of a Logarithm: A solid understanding of what a logarithm represents is fundamental. Remember, $\log_b(x) = y$ is equivalent to $b^y = x$. If you're unsure about this relationship, revisit the definition and practice converting between logarithmic and exponential forms.

By being mindful of these common mistakes, you can significantly improve your accuracy and confidence in solving logarithmic equations. So, keep these points in mind, double-check your work, and you'll be well on your way to mastering logarithms!

Practice Problems

Okay, now that we've gone through a detailed example and highlighted common mistakes, it's time to put your skills to the test! Practice is key to mastering any mathematical concept, and logarithmic equations are no exception. Here are a few problems for you to tackle. Work through them step by step, remembering to check your solutions for extraneous roots. Good luck, and have fun!

1. $\log_2(x + 3) + \log_2(x - 1) = 2$
2. $2\log(x) = \log(2x + 3)$
3. $\log_3(x^2 - 4) - \log_3(x + 2) = 1$
4. $\log_5(2x - 1) = 2$
5. $\log(x + 4) - \log(x - 2) = \log(5)$

Hint: Remember to use the properties of logarithms to combine or simplify the equations before converting them to exponential form. And don't forget to check those solutions! 🕵️‍♀️

Solving these practice problems will not only reinforce your understanding of the steps involved but also help you develop the intuition needed to approach different types of logarithmic equations. So, grab a pen and paper, and let's get practicing! You've got this!

Conclusion

Alright guys, we've reached the end of our logarithmic equation-solving journey! We've covered a lot of ground, from understanding the basic principles and properties of logarithms to working through a detailed example and highlighting common mistakes. Remember, solving logarithmic equations is all about breaking the problem down into manageable steps: rearranging, combining logarithms, converting to exponential form, solving for the variable, and, crucially, checking your solutions.

The key to mastering these equations is practice, practice, practice! The more you work with logarithms, the more comfortable and confident you'll become. Don't be discouraged if you encounter tricky problems along the way. Instead, see them as opportunities to sharpen your skills and deepen your understanding.

So, keep those logarithmic properties handy, watch out for those common mistakes, and keep practicing. You'll be solving logarithmic equations like a pro in no time! Happy solving, and remember, math can be fun! 😄