Solving Logarithmic Equations: A Step-by-Step Guide

Oct 25, 2025 by ADMIN 52 views

Hey guys! Today, we're diving into the exciting world of logarithmic equations. Logarithmic equations might seem intimidating at first, but don't worry, we'll break it down step by step. We'll tackle a specific problem, but the principles we cover can be applied to a wide range of logarithmic equations. So, let's jump right in and learn how to solve logarithmic equations like pros!

Understanding the Logarithmic Equation

Before we jump into solving, let's understand the logarithmic equation we're dealing with. We have:

log₅(x) + log₅(4x - 1) = 1

This equation involves logarithms with the same base (base 5), which is excellent news for us because it allows us to use some handy logarithmic properties. The goal here is to find the value(s) of x that satisfy this equation. Remember, x needs to be within the domain of the logarithmic expressions, meaning the arguments of the logarithms (x and 4x - 1) must be greater than zero. This is a crucial step to consider, as we'll need to check our solutions later.

Domain Restrictions: A Critical First Step

Before we even start manipulating the equation, it's essential to identify any domain restrictions. Remember, you can't take the logarithm of a non-positive number (zero or negative). Therefore, we have two conditions to satisfy:

x > 0
4x - 1 > 0

The first condition, x > 0, is straightforward. For the second condition, let's solve for x:

4x - 1 > 0

Add 1 to both sides:

4x > 1

Divide by 4:

x > 1/4

So, we have two restrictions: x must be greater than 0 and greater than 1/4. Combining these, we see that x > 1/4 is our overall domain restriction. Any solution we find must satisfy this condition. Keep this in mind, guys; it's a common mistake to forget about the domain restrictions and end up with extraneous solutions!

Step 1: Condensing the Logarithmic Expression

The key to solving many logarithmic equations is to use the properties of logarithms to condense the expression. In this case, we can use the product rule of logarithms, which states:

logₐ(m) + logₐ(n) = logₐ(m * n)

Applying this rule to our equation, we get:

log₅(x) + log₅(4x - 1) = log₅(x * (4x - 1))

So our equation now looks like this:

log₅(x(4x - 1)) = 1

This is progress! We've combined the two logarithms into a single logarithm. The next step is to rewrite the equation in exponential form.

Step 2: Rewriting in Exponential Form

Now, let's rewrite the logarithmic equation in its equivalent exponential form. Remember the basic relationship between logarithms and exponents:

logₐ(b) = c is equivalent to aᶜ = b

In our case, we have log₅(x(4x - 1)) = 1. Applying the definition, we get:

5¹ = x(4x - 1)

This simplifies to:

5 = x(4x - 1)

Great! We've eliminated the logarithm, and we're now dealing with a simple algebraic equation. This is much easier to handle.

Step 3: Solving the Quadratic Equation

Let's simplify and rearrange the equation to get a standard quadratic equation. Distribute the x on the right side:

5 = 4x² - x

Now, subtract 5 from both sides to set the equation equal to zero:

4x² - x - 5 = 0

We now have a quadratic equation in the form ax² + bx + c = 0. There are several ways to solve quadratic equations, including factoring, using the quadratic formula, or completing the square. In this case, let's try factoring. We need to find two numbers that multiply to (4 * -5) = -20 and add up to -1. Those numbers are -5 and 4.

We can rewrite the middle term using these numbers:

4x² - 5x + 4x - 5 = 0

Now, factor by grouping:

x(4x - 5) + 1(4x - 5) = 0

Factor out the common factor (4x - 5):

(4x - 5)(x + 1) = 0

Now, set each factor equal to zero and solve for x:

4x - 5 = 0 or x + 1 = 0

Solving these, we get:

4x = 5 => x = 5/4

x = -1

So, we have two potential solutions: x = 5/4 and x = -1. But remember, we need to check these solutions against our domain restriction!

Step 4: Checking for Extraneous Solutions

This is a critical step that you should never skip when solving logarithmic equations. We need to make sure our solutions don't violate the domain restriction we found earlier (x > 1/4). Let's check our solutions:

x = 5/4: This is greater than 1/4, so it satisfies our domain restriction. This is a valid solution.
x = -1: This is less than 1/4 (and also negative), so it violates our domain restriction. This is an extraneous solution and we must reject it.

Extraneous solutions often arise when dealing with logarithmic and radical equations, so always remember to check your answers!

Step 5: The Final Solution

After checking for extraneous solutions, we can confidently state our solution. The only solution that satisfies the original equation and the domain restriction is:

x = 5/4

Therefore, the solution to the logarithmic equation log₅(x) + log₅(4x - 1) = 1 is x = 5/4. We solved it! Remember, the key is to condense the logarithmic expression, rewrite it in exponential form, solve the resulting equation, and, most importantly, check for extraneous solutions. You've got this, guys!

Key Takeaways for Solving Logarithmic Equations

To wrap things up, let's recap the crucial steps involved in solving logarithmic equations. These steps will help you tackle similar problems with confidence:

Identify Domain Restrictions: Before you start manipulating the equation, determine the values of x that make the arguments of the logarithms positive. This is crucial for avoiding extraneous solutions.
Condense Logarithmic Expressions: Use the properties of logarithms (product rule, quotient rule, power rule) to combine multiple logarithms into a single logarithm. This simplifies the equation and makes it easier to solve.
Rewrite in Exponential Form: Convert the logarithmic equation into its equivalent exponential form. This eliminates the logarithms and transforms the equation into an algebraic one.
Solve the Algebraic Equation: Solve the resulting algebraic equation. This might involve solving a linear equation, a quadratic equation, or another type of equation. Use appropriate techniques such as factoring, the quadratic formula, or other algebraic methods.
Check for Extraneous Solutions: This is a critical step. Substitute your solutions back into the original logarithmic equation and verify that they satisfy the domain restrictions. Reject any solutions that make the argument of a logarithm non-positive.
State the Solution: After checking for extraneous solutions, clearly state the final solution(s) to the logarithmic equation.

By following these steps, you'll be well-equipped to solve a wide variety of logarithmic equations. Remember, practice makes perfect, so keep working through examples and building your skills. You'll become a pro in no time! And most importantly, don't forget to have fun while learning!

Practice Problems to Master Logarithmic Equations

Want to put your newfound skills to the test? Here are a few practice problems to help you master solving logarithmic equations:

log₂(x) + log₂(x - 2) = 3
log₃(2x + 1) - log₃(x - 2) = 2
2log₅(x) = log₅(9)

Work through these problems, applying the steps we discussed. Remember to check for extraneous solutions! The more you practice, the more comfortable you'll become with logarithmic equations.

Solving logarithmic equations is a valuable skill in mathematics. By understanding the properties of logarithms, applying the correct techniques, and remembering to check for extraneous solutions, you can confidently tackle these types of problems. Keep practicing, and you'll be a logarithmic equation-solving master in no time!