Solving Logarithmic Equations A Step By Step Guide With Example

Hey there, math enthusiasts! Today, we're diving deep into the world of logarithmic equations. Logarithmic equations might seem intimidating at first, but with a systematic approach and a clear understanding of the underlying principles, you'll be solving them like a pro in no time. In this comprehensive guide, we'll break down the steps involved in solving logarithmic equations, emphasizing the crucial step of rejecting extraneous solutions. We'll also explore the concept of rewriting logarithmic equations in exponential form and the importance of the domain of logarithmic functions. So, buckle up and get ready to conquer those logs!

Understanding Logarithmic Equations

Before we jump into solving logarithmic equations, let's quickly recap what logarithms are all about. Logarithms are essentially the inverse operation of exponentiation. Think of it this way: if $b^y = x$, then we can write this in logarithmic form as $\log_b x = y$. Here, b is the base of the logarithm, x is the argument (the number we're taking the logarithm of), and y is the exponent to which we must raise b to get x. In simpler terms, the logarithm tells you what power you need to raise the base to in order to get a specific number.

Now, what exactly is a logarithmic equation? A logarithmic equation is simply an equation that involves one or more logarithms. Our goal when solving these equations is to isolate the variable, just like with any other type of equation. However, because of the nature of logarithms, we need to be extra careful about the solutions we obtain. This is where the concept of the domain of a logarithmic function comes into play.

The domain of a logarithmic function is the set of all possible input values (arguments) for which the function is defined. Remember, you can only take the logarithm of a positive number. This means that the argument of a logarithm must always be greater than zero. This restriction is crucial when solving logarithmic equations because it can lead to extraneous solutions – values that satisfy the transformed equation but not the original logarithmic equation.

Key Concepts to Remember

• Logarithm as the inverse of exponentiation: If $b^y = x$, then $\log_b x = y$.
• Base of the logarithm: The base b must be a positive number not equal to 1.
• Argument of the logarithm: The argument x must be a positive number.
• Domain of a logarithmic function: The set of all positive real numbers.
• Extraneous solutions: Solutions that satisfy the transformed equation but not the original logarithmic equation.

Solving the Logarithmic Equation $5 \ln(2x) = 15$

Let's tackle the specific logarithmic equation you provided: $5 \ln(2x) = 15$. We'll walk through the steps systematically, highlighting the importance of checking for extraneous solutions.

Step 1: Isolate the Logarithmic Term

Our first goal is to isolate the logarithmic term, which in this case is $\ln(2x)$. To do this, we need to get rid of the coefficient 5. We can achieve this by dividing both sides of the equation by 5:

$5 \ln(2x) / 5 = 15 / 5$

This simplifies to:

$\ln(2x) = 3$

Great! We've successfully isolated the logarithmic term.

Step 2: Rewrite in Exponential Form

Now comes the crucial step of rewriting the equation in exponential form. Remember that the natural logarithm, denoted by $\ln$, has a base of e (Euler's number, approximately 2.71828). So, the equation $\ln(2x) = 3$ is equivalent to:

$e^3 = 2x$

We've successfully transformed the logarithmic equation into an exponential equation. This is a key step in solving for x.

Step 3: Solve for $x$

To solve for x, we simply need to isolate it. In this case, we can do this by dividing both sides of the equation by 2:

$e^3 / 2 = 2x / 2$

This simplifies to:

$x = e^3 / 2$

So, we've found a potential solution for x. But hold on! We're not done yet. We need to check for extraneous solutions.

Step 4: Check for Extraneous Solutions

This is the most important step! Remember that the argument of a logarithm must be positive. In our original equation, the argument is $2x$. So, we need to make sure that $2x > 0$. Let's substitute our potential solution, $x = e^3 / 2$, into this inequality:

$2(e^3 / 2) > 0$

This simplifies to:

$e^3 > 0$

Since $e^3$ is approximately 20.0855, which is indeed greater than 0, our solution $x = e^3 / 2$ satisfies the domain restriction. Therefore, it is a valid solution.

The Solution

The solution to the logarithmic equation $5 \ln(2x) = 15$ is $x = e^3 / 2$.

Rewriting the Equation Without Logarithms

As part of the original problem, you were asked to rewrite the given equation without logarithms. We actually did this in Step 2! When we transformed the equation $\ln(2x) = 3$ into its exponential form, we got:

$e^3 = 2x$

This is the equation rewritten without logarithms. We haven't solved for x yet in this form, but we've successfully eliminated the logarithmic term.

Discussion on the Domain of Logarithmic Functions

The domain of a function is a critical concept in mathematics, and it's especially important when dealing with logarithmic functions. As we've emphasized, the argument of a logarithm must be positive. This restriction stems from the definition of logarithms as the inverse of exponentiation.

Consider the exponential function $b^y = x$. If b is a positive number (and not equal to 1), then no matter what value we choose for y, the result x will always be positive. This is because a positive number raised to any power (positive, negative, or zero) will always be positive. Therefore, when we take the logarithm of x (i.e., $\log_b x = y$), x must be positive.

Why is the Domain Restriction Important?

Failing to consider the domain restriction can lead to extraneous solutions, which are incorrect answers. These solutions might satisfy the transformed equation after you've manipulated it, but they don't work in the original logarithmic equation because they would result in taking the logarithm of a non-positive number.

For example, let's say we encountered a solution that made the argument of the logarithm negative. If we plugged that value back into the original equation, we'd be trying to take the logarithm of a negative number, which is undefined. This is a clear indication that the solution is extraneous and must be rejected.

How to Determine the Domain

To determine the domain of a logarithmic function, simply set the argument of the logarithm greater than zero and solve for x. For example, in the equation $5 \ln(2x) = 15$, the argument is $2x$. So, we would set $2x > 0$ and solve for x, which gives us $x > 0$. This means that the domain of the function is all positive real numbers.

Common Mistakes to Avoid

Solving logarithmic equations can be tricky, and there are a few common mistakes that students often make. Let's highlight these pitfalls so you can steer clear of them:

1. Forgetting to Check for Extraneous Solutions: This is the biggest mistake! Always, always, always check your solutions by plugging them back into the original equation to make sure they don't violate the domain restriction.
2. Incorrectly Rewriting in Exponential Form: Make sure you understand the relationship between logarithmic and exponential forms. If you're not sure, write out the definition ($b^y = x$ is equivalent to $\log_b x = y$) and use it as a guide.
3. Misunderstanding the Properties of Logarithms: Logarithms have several important properties that can be helpful in solving equations. Make sure you're familiar with these properties and how to apply them correctly. Some key properties include the product rule, quotient rule, and power rule.
4. Not Isolating the Logarithmic Term First: Before you can rewrite the equation in exponential form, you need to isolate the logarithmic term. This usually involves using algebraic operations like addition, subtraction, multiplication, or division to get the logarithm by itself on one side of the equation.

Practice Makes Perfect

The best way to master solving logarithmic equations is to practice, practice, practice! Work through a variety of examples, paying close attention to the steps involved and the importance of checking for extraneous solutions. The more you practice, the more comfortable and confident you'll become.

Conclusion

Solving logarithmic equations is a fundamental skill in mathematics. By understanding the relationship between logarithms and exponents, the domain restrictions of logarithmic functions, and the importance of checking for extraneous solutions, you can confidently tackle these equations. Remember to isolate the logarithmic term, rewrite in exponential form, solve for the variable, and always check your answers. With practice and a systematic approach, you'll become a logarithmic equation-solving whiz!

So, go forth and conquer those logs! You've got this!