Solve The Inequality:$\log _3\left(x^2-3 X-67\right)\ \textless \ 3$

$$

Introduction

In this article, we will focus on solving the given inequality $\log _3\left(x^2-3 x-67\right)\ \textless \ 3$. This involves understanding the properties of logarithms and how to manipulate them to isolate the variable. We will break down the solution into manageable steps, making it easier to follow and understand.

Understanding the Inequality

The given inequality is $\log _3\left(x^2-3 x-67\right)\ \textless \ 3$. To solve this, we need to first understand the properties of logarithms. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number. In this case, we have a logarithm with base 3.

Step 1: Exponentiate Both Sides

To solve the inequality, we can start by exponentiating both sides. This will help us get rid of the logarithm and make it easier to work with. When we exponentiate both sides, we get:

$$3^{\log _3\left(x^2-3 x-67\right)}\ \textless \ 3^3$$

Step 2: Simplify the Left Side

The left side of the inequality can be simplified using the property of logarithms that states $\log _a\left(a^x\right)=x$. Applying this property, we get:

$$x^2-3 x-67\ \textless \ 27$$

Step 3: Rearrange the Inequality

To make it easier to work with, we can rearrange the inequality by subtracting 27 from both sides. This gives us:

$$x^2-3 x-94\ \textless \ 0$$

Step 4: Factor the Quadratic Expression

The quadratic expression $x^2-3 x-94$ can be factored as:

$$(x-17)(x+5.5)\ \textless \ 0$$

Step 5: Find the Critical Points

To find the critical points, we need to set each factor equal to zero and solve for x. This gives us:

$$x-17=0\Rightarrow x=17$$

$$x+5.5=0\Rightarrow x=-5.5$$

Step 6: Test the Intervals

To determine the solution to the inequality, we need to test the intervals between the critical points. We can do this by plugging in a test value from each interval into the original inequality.

Step 7: Determine the Solution

After testing the intervals, we find that the solution to the inequality is:

$$-5.5\ \textless \ x\ \textless \ 17$$

Conclusion

In this article, we solved the inequality $\log _3\left(x^2-3 x-67\right)\ \textless \ 3$ by exponentiating both sides, simplifying the left side, rearranging the inequality, factoring the quadratic expression, finding the critical points, testing the intervals, and determining the solution. The final solution is $-5.5\ \textless \ x\ \textless \ 17$.

Additional Tips and Tricks

• When solving inequalities, it's essential to remember that the direction of the inequality sign may change when multiplying or dividing both sides by a negative number.
• When factoring quadratic expressions, it's crucial to find the critical points and test the intervals to determine the solution.
• When working with logarithms, it's essential to remember the properties of logarithms and how to manipulate them to isolate the variable.

Frequently Asked Questions

• Q: What is the solution to the inequality $\log _3\left(x^2-3 x-67\right)\ \textless \ 3$? A: The solution to the inequality is $-5.5\ \textless \ x\ \textless \ 17$.
• Q: How do I solve inequalities with logarithms? A: To solve inequalities with logarithms, you can start by exponentiating both sides, simplifying the left side, rearranging the inequality, factoring the quadratic expression, finding the critical points, testing the intervals, and determining the solution.

Final Thoughts

Solving inequalities with logarithms requires a deep understanding of the properties of logarithms and how to manipulate them to isolate the variable. By following the steps outlined in this article, you can solve inequalities with logarithms and gain a deeper understanding of the subject.

Introduction

In the previous article, we solved the inequality $\log _3\left(x^2-3 x-67\right)\ \textless \ 3$ by exponentiating both sides, simplifying the left side, rearranging the inequality, factoring the quadratic expression, finding the critical points, testing the intervals, and determining the solution. In this article, we will answer some of the most frequently asked questions about solving inequalities with logarithms.

Q&A

Q: What is the difference between solving inequalities with logarithms and solving equations with logarithms?

A: Solving inequalities with logarithms involves finding the values of x that satisfy the inequality, whereas solving equations with logarithms involves finding the values of x that satisfy the equation. In other words, inequalities have a "less than" or "greater than" sign, whereas equations have an "=" sign.

Q: How do I know which base to use when solving inequalities with logarithms?

A: The base of the logarithm is usually given in the problem. If it's not given, you can choose any base you like, but it's usually easiest to use the base that is most convenient for the problem.

Q: Can I use the same steps to solve inequalities with logarithms as I would to solve equations with logarithms?

A: No, the steps are different. When solving inequalities with logarithms, you need to exponentiate both sides, simplify the left side, rearrange the inequality, factor the quadratic expression, find the critical points, test the intervals, and determine the solution. When solving equations with logarithms, you can simply use the property of logarithms that states $\log _a\left(a^x\right)=x$.

Q: What if the inequality has a negative number in the base of the logarithm?

A: If the inequality has a negative number in the base of the logarithm, you can use the property of logarithms that states $\log _a\left(a^x\right)=x$ to simplify the left side. However, you need to be careful when exponentiating both sides, as the direction of the inequality sign may change.

Q: Can I use a calculator to solve inequalities with logarithms?

A: Yes, you can use a calculator to solve inequalities with logarithms. 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 hand to make sure you get the correct answer.

Q: What if the inequality has a fractional exponent?

A: If the inequality has a fractional exponent, you can use the property of logarithms that states $\log _a\left(a^x\right)=x$ to simplify the left side. However, you need to be careful when exponentiating both sides, as the direction of the inequality sign may change.

Q: Can I use the same steps to solve inequalities with logarithms as I would to solve inequalities with exponents?

A: No, the steps are different. When solving inequalities with logarithms, you need to exponentiate both sides, simplify the left side, rearrange the inequality, factor the quadratic expression, find the critical points, test the intervals, and determine the solution. When solving inequalities with exponents, you can simply use the property of exponents that states $a^x\cdot a^y=a^{x+y}$.

Conclusion

Solving inequalities with logarithms can be a challenging task, but with the right steps and a little practice, you can become proficient in solving them. Remember to always exponentiate both sides, simplify the left side, rearrange the inequality, factor the quadratic expression, find the critical points, test the intervals, and determine the solution. With these steps and a little practice, you can become a master of solving inequalities with logarithms.

Additional Tips and Tricks

• When solving inequalities with logarithms, it's essential to remember the properties of logarithms and how to manipulate them to isolate the variable.
• When working with logarithms, it's crucial to remember the base of the logarithm and how to use it to simplify the left side.
• When exponentiating both sides, be careful when the direction of the inequality sign may change.
• When using a calculator to solve inequalities with logarithms, always check your work by hand to make sure you get the correct answer.

Final Thoughts

Solving inequalities with logarithms requires a deep understanding of the properties of logarithms and how to manipulate them to isolate the variable. By following the steps outlined in this article and practicing regularly, you can become proficient in solving inequalities with logarithms and gain a deeper understanding of the subject.