Which Of The Following Logarithmic Equations Is Equivalent To $x^3=\frac{1}{27}$?A. $\log _3 \frac{1}{27}=x$B. $\log _x \frac{1}{27}=3$C. $\log _x 27=3$

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will explore which of the given logarithmic equations is equivalent to the equation $x^3=\frac{1}{27}$. We will break down each option and provide a step-by-step solution to determine the correct answer.

Understanding the Given Equation

The given equation is $x^3=\frac{1}{27}$. To solve this equation, we need to find the value of $x$ that satisfies the equation. We can start by rewriting the equation in a more familiar form.

$$x^3=\frac{1}{27}$$

We can rewrite $\frac{1}{27}$ as $3^{-3}$, since $27=3^3$.

$$x^3=3^{-3}$$

Now, we can take the cube root of both sides of the equation to solve for $x$.

$$x=\sqrt[3]{3^{-3}}$$

Using the property of exponents, we can simplify the expression.

$$x=3^{-1}$$

$$x=\frac{1}{3}$$

Analyzing Option A

Option A is $\log _3 \frac{1}{27}=x$. To determine if this option is equivalent to the given equation, we need to evaluate the logarithmic expression.

$$\log _3 \frac{1}{27}$$

We can rewrite $\frac{1}{27}$ as $3^{-3}$, since $27=3^3$.

$$\log _3 3^{-3}$$

Using the property of logarithms, we can simplify the expression.

$$-3\log _3 3$$

Since $\log _3 3=1$, we can further simplify the expression.

$$-3$$

This is not equal to the value of $x$ we found earlier, which is $\frac{1}{3}$. Therefore, option A is not equivalent to the given equation.

Analyzing Option B

Option B is $\log _x \frac{1}{27}=3$. To determine if this option is equivalent to the given equation, we need to evaluate the logarithmic expression.

$$\log _x \frac{1}{27}=3$$

We can rewrite $\frac{1}{27}$ as $3^{-3}$, since $27=3^3$.

$$\log _x 3^{-3}=3$$

Using the property of logarithms, we can simplify the expression.

$$-3\log _x 3=3$$

Since $\log _x 3$ is the inverse of $x^3$, we can rewrite the expression as.

$$-3\frac{1}{x^3}=3$$

Multiplying both sides of the equation by $-x^3$, we get.

$$3x^3=-9$$

Dividing both sides of the equation by $3$, we get.

$$x^3=-3$$

This is not equal to the value of $x$ we found earlier, which is $\frac{1}{3}$. Therefore, option B is not equivalent to the given equation.

Analyzing Option C

Option C is $\log _x 27=3$. To determine if this option is equivalent to the given equation, we need to evaluate the logarithmic expression.

$$\log _x 27=3$$

We can rewrite $27$ as $3^3$, since $27=3^3$.

$$\log _x 3^3=3$$

Using the property of logarithms, we can simplify the expression.

$$3\log _x 3=3$$

Dividing both sides of the equation by $3$, we get.

$$\log _x 3=1$$

This means that $x^3=3^1$, which is equivalent to $x^3=3$. Therefore, option C is equivalent to the given equation.

Conclusion

$$$$$$$$

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will provide a Q&A guide to help you understand logarithmic equations and how to solve them.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse of an exponential function. Logarithmic equations are used to solve problems that involve exponential growth or decay.

Q: What are the properties of logarithms?

A: The properties of logarithms are:

• Product rule: $\log _a (xy)=\log _a x + \log _a y$
• Quotient rule: $\log _a \frac{x}{y}=\log _a x - \log _a y$
• Power rule: $\log _a x^y=y\log _a x$
• Change of base rule: $\log _a x=\frac{\log _b x}{\log _b a}$

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to follow these steps:

1. Isolate the logarithm: Move all terms except the logarithm to one side of the equation.
2. Use the properties of logarithms: Use the product rule, quotient rule, power rule, or change of base rule to simplify the equation.
3. Exponentiate both sides: Raise both sides of the equation to the power of the base of the logarithm.
4. Solve for the variable: Solve for the variable in the equation.

Q: What is the difference between a logarithmic equation and an exponential equation?

A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponential function. For example:

• Logarithmic equation: $\log _a x=y$
• Exponential equation: $a^y=x$

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

A: Yes, you can use a calculator to solve logarithmic equations. Most calculators have a logarithm function that allows you to input a base and a value, and it will return the logarithm of the value.

Q: What are some common mistakes to avoid when solving logarithmic equations?

A: Some common mistakes to avoid when solving logarithmic equations include:

• Not isolating the logarithm: Failing to isolate the logarithm can make it difficult to solve the equation.
• Not using the properties of logarithms: Failing to use the properties of logarithms can make it difficult to simplify the equation.
• Not exponentiating both sides: Failing to exponentiate both sides of the equation can lead to incorrect solutions.

Q: How do I check my answer when solving a logarithmic equation?

A: To check your answer when solving a logarithmic equation, you can:

• Plug in the solution: Plug the solution back into the original equation to see if it is true.
• Use a calculator: Use a calculator to check if the solution is correct.
• Graph the equation: Graph the equation to see if the solution is a valid solution.

Conclusion

In conclusion, logarithmic equations are an important concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. By following the steps outlined in this article, you can solve logarithmic equations and check your answers. Remember to avoid common mistakes and use a calculator or graphing tool to check your solutions.