Key Property of Logarithms

Oct 13, 2025 by leolavanderia.com.br 71 views

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Introduction

Logarithmic equations can be challenging to solve, but by breaking them down into manageable steps, we can make the process more accessible. In this article, we will guide you through the steps to solve the equation $\log(x^2 - 15) = \log(2x)$ , from 1 to 5. By following these steps, you will be able to tackle similar logarithmic equations with confidence.

Step 1: Understand the Equation

The given equation is $\log(x^2 - 15) = \log(2x)$ . To begin solving this equation, we need to understand the properties of logarithms. The logarithm of a number is the exponent to which a base number must be raised to produce that number. In this case, we have two logarithmic expressions on both sides of the equation.

Key Property of Logarithms

One of the key properties of logarithms is that if $\log(a) = \log(b)$ , then $a = b$ . This property will be essential in solving the equation.

Step 2: Eliminate the Logarithms

Using the property mentioned above, we can eliminate the logarithms from the equation. Since $\log(x^2 - 15) = \log(2x)$ , we can equate the expressions inside the logarithms:

$x^2 - 15 = 2x$

Simplifying the Equation

Now, we can simplify the equation by moving all terms to one side:

$x^2 - 2x - 15 = 0$

Step 3: Factor the Quadratic Equation

The equation $x^2 - 2x - 15 = 0$ is a quadratic equation. We can factor it to find the values of $x$ that satisfy the equation:

$(x - 5)(x + 3) = 0$

Solving for $x$

To find the values of $x$ , we can set each factor equal to zero and solve for $x$ :

$x - 5 = 0 \Rightarrow x = 5$

$x + 3 = 0 \Rightarrow x = -3$

Step 4: Check the Solutions

Before accepting the solutions, we need to check if they satisfy the original equation. We can plug in each solution into the original equation to verify its validity.

Checking $x = 5$

$\log(5^2 - 15) = \log(2(5))$

$\log(20) = \log(10)$

Since $\log(20) \neq \log(10)$ , $x = 5$ is not a valid solution.

Checking $x = -3$

$\log((-3)^2 - 15) = \log(2(-3))$

$\log(0) = \log(-6)$

Since $\log(0)$ is undefined, $x = -3$ is not a valid solution.

Step 5: Conclusion

In this article, we have walked through the steps to solve the equation $\log(x^2 - 15) = \log(2x)$ , from 1 to 5. We have used the properties of logarithms, eliminated the logarithms, factored the quadratic equation, and checked the solutions. Unfortunately, neither of the solutions satisfied the original equation. This highlights the importance of carefully checking the solutions to ensure their validity.

Takeaway

Solving logarithmic equations requires a step-by-step approach. By understanding the properties of logarithms, eliminating the logarithms, factoring the quadratic equation, and checking the solutions, we can tackle even the most challenging logarithmic equations. Remember to always verify the solutions to ensure their validity.

Final Thoughts

Logarithmic equations may seem daunting at first, but with practice and patience, you can become proficient in solving them. By following the steps outlined in this article, you will be well on your way to mastering logarithmic equations.

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Introduction

In our previous article, we walked through the steps to solve the equation $\log(x^2 - 15) = \log(2x)$ , from 1 to 5. However, we encountered some challenges along the way. In this article, we will address some of the most frequently asked questions related to solving logarithmic equations.

Q: What is the first step in solving a logarithmic equation?

A: The first step in solving a logarithmic equation is to understand the properties of logarithms. Specifically, you need to know that if $\log(a) = \log(b)$ , then $a = b$ . This property will help you eliminate the logarithms from the equation.

Q: How do I eliminate the logarithms from the equation?

A: To eliminate the logarithms, you can equate the expressions inside the logarithms. For example, if $\log(x^2 - 15) = \log(2x)$ , you can write $x^2 - 15 = 2x$ . This will help you simplify the equation and make it easier to solve.

Q: What if the equation is a quadratic equation? How do I solve it?

A: If the equation is a quadratic equation, you can try factoring it. For example, if the equation is $x^2 - 2x - 15 = 0$ , you can factor it as $(x - 5)(x + 3) = 0$ . This will help you find the values of $x$ that satisfy the equation.

Q: How do I check the solutions to ensure their validity?

A: To check the solutions, you need to plug them back into the original equation. For example, if you found that $x = 5$ is a solution, you need to check if $\log(5^2 - 15) = \log(2(5))$ is true. If it's not, then $x = 5$ is not a valid solution.

Q: What if I encounter a logarithm with a negative number inside? How do I handle it?

A: If you encounter a logarithm with a negative number inside, you need to be careful. The logarithm of a negative number is undefined, so you need to find a way to eliminate the negative number. For example, if you have $\log(-x)$ , you can rewrite it as $\log(-1) + \log(x)$ .

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

A: Yes, you can use a calculator to solve logarithmic equations. However, you need to be careful when using a calculator, as it may not always give you the correct answer. It's always a good idea to check your work by plugging the solution back into the original equation.

Q: How do I know if a logarithmic equation has a solution?

A: To determine if a logarithmic equation has a solution, you need to check if the equation is true for any value of $x$ . If the equation is true for any value of $x$ , then it has a solution. If the equation is not true for any value of $x$ , then it does not have a solution.

Q: Can I use logarithmic equations to solve real-world problems?

A: Yes, you can use logarithmic equations to solve real-world problems. Logarithmic equations are used in many fields, including science, engineering, and finance. For example, you can use logarithmic equations to model population growth, chemical reactions, and financial transactions.

Conclusion

Solving logarithmic equations requires a step-by-step approach. By understanding the properties of logarithms, eliminating the logarithms, factoring the quadratic equation, and checking the solutions, you can tackle even the most challenging logarithmic equations. Remember to always verify the solutions to ensure their validity.