Use The Like Bases Property To Solve The Following Equation.${ 3^n = 27 }$Enter Your Answer For { N $}$:

Mar 9, 2025 by ADMIN 106 views

**Solving Exponential Equations with the Logarithmic Base Property**

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of logarithmic properties. In this article, we will explore how to use the logarithmic base property to solve the equation $3^n = 27$ . This equation is a classic example of an exponential equation, and solving it will require us to apply the logarithmic base property.

Understanding Exponential Equations

Exponential equations are equations that involve an exponential expression, which is an expression of the form $a^x$ , where $a$ is a positive number and $x$ is a variable. In the equation $3^n = 27$ , the base is $3$ and the exponent is $n$ . The equation states that $3$ raised to the power of $n$ is equal to $27$ .

The Logarithmic Base Property

The logarithmic base property is a fundamental concept in mathematics that states that if $a^x = b$ , then $x = \log_a b$ . This property allows us to solve exponential equations by taking the logarithm of both sides of the equation. In the case of the equation $3^n = 27$ , we can take the logarithm of both sides to get $n = \log_3 27$ .

Solving the Equation

To solve the equation $3^n = 27$ , we can use the logarithmic base property. We start by taking the logarithm of both sides of the equation:

\log_3 3^n = \log_3 27

Using the property of logarithms that states $\log_a a^x = x$ , we can simplify the left-hand side of the equation to get:

n = \log_3 27

Finding the Value of n

To find the value of $n$ , we need to evaluate the logarithm $\log_3 27$ . We can do this by using the change of base formula, which states that $\log_a b = \frac{\log_c b}{\log_c a}$ , where $c$ is any positive number. In this case, we can choose $c = 10$ :

n = \frac{\log_{10} 27}{\log_{10} 3}

Evaluating the Logarithms

To evaluate the logarithms, we can use a calculator or a logarithm table. Using a calculator, we get:

\log_{10} 27 \approx 1.4313

\log_{10} 3 \approx 0.4771

Finding the Value of n

Now that we have the values of the logarithms, we can substitute them into the equation to get:

n \approx \frac{1.4313}{0.4771}

Simplifying the Expression

To simplify the expression, we can divide the numerator by the denominator:

n \approx 3

Conclusion

In this article, we have shown how to use the logarithmic base property to solve the equation $3^n = 27$ . We started by taking the logarithm of both sides of the equation and then used the change of base formula to evaluate the logarithm. Finally, we simplified the expression to get the value of $n$ . The value of $n$ is approximately $3$ , which means that $3^3 = 27$ .

Real-World Applications

Exponential equations have many real-world applications, including finance, science, and engineering. For example, in finance, exponential equations are used to model the growth of investments and the decay of assets. In science, exponential equations are used to model the growth and decay of populations, the spread of diseases, and the behavior of physical systems. In engineering, exponential equations are used to model the behavior of electronic circuits, the growth of materials, and the decay of radioactive substances.

Tips and Tricks

When solving exponential equations, it is essential to remember the following tips and tricks:

Use the logarithmic base property: The logarithmic base property is a powerful tool for solving exponential equations. It allows us to take the logarithm of both sides of the equation and then use the change of base formula to evaluate the logarithm.
Choose the right base: When using the logarithmic base property, it is essential to choose the right base. In this case, we chose the base $3$ because it is the base of the exponential expression.
Use a calculator or logarithm table: When evaluating logarithms, it is essential to use a calculator or logarithm table. This will ensure that you get accurate results.
Simplify the expression: Finally, it is essential to simplify the expression to get the value of $n$ . This will ensure that you get the correct answer.

Conclusion

In conclusion, solving exponential equations with the logarithmic base property is a powerful tool for solving equations that involve exponential expressions. By using the logarithmic base property, we can take the logarithm of both sides of the equation and then use the change of base formula to evaluate the logarithm. This will give us the value of $n$ , which is the exponent of the exponential expression.

Q: What is the logarithmic base property?

A: The logarithmic base property is a fundamental concept in mathematics that states that if $a^x = b$ , then $x = \log_a b$ . This property allows us to solve exponential equations by taking the logarithm of both sides of the equation.

Q: How do I use the logarithmic base property to solve an exponential equation?

A: To use the logarithmic base property to solve an exponential equation, you need to take the logarithm of both sides of the equation. This will give you an equation in the form $x = \log_a b$ , where $x$ is the exponent of the exponential expression.

Q: What is the change of base formula?

A: The change of base formula is a formula that allows us to change the base of a logarithm. It states that $\log_a b = \frac{\log_c b}{\log_c a}$ , where $c$ is any positive number.

Q: How do I choose the right base for the logarithmic base property?

A: When using the logarithmic base property, it is essential to choose the right base. In general, you should choose a base that is easy to work with and that is related to the exponential expression. For example, if the exponential expression is $3^n$ , it is a good idea to choose the base $3$ .

Q: What are some common mistakes to avoid when using the logarithmic base property?

A: Some common mistakes to avoid when using the logarithmic base property include:

Not taking the logarithm of both sides of the equation: This will give you an incorrect solution.
Not using the change of base formula: This will give you an incorrect solution.
Not choosing the right base: This will give you an incorrect solution.

Q: Can I use the logarithmic base property to solve exponential equations with negative exponents?

A: Yes, you can use the logarithmic base property to solve exponential equations with negative exponents. However, you need to be careful when taking the logarithm of both sides of the equation, as this will give you a negative exponent.

Q: Can I use the logarithmic base property to solve exponential equations with fractional exponents?

A: Yes, you can use the logarithmic base property to solve exponential equations with fractional exponents. However, you need to be careful when taking the logarithm of both sides of the equation, as this will give you a fractional exponent.

Q: What are some real-world applications of the logarithmic base property?

A: The logarithmic base property has many real-world applications, including:

Finance: Exponential equations are used to model the growth of investments and the decay of assets.
Science: Exponential equations are used to model the growth and decay of populations, the spread of diseases, and the behavior of physical systems.
Engineering: Exponential equations are used to model the behavior of electronic circuits, the growth of materials, and the decay of radioactive substances.

Q: How do I simplify the expression after using the logarithmic base property?

A: To simplify the expression after using the logarithmic base property, you need to use the properties of logarithms, such as the product rule and the quotient rule. You also need to use the change of base formula to evaluate the logarithm.

Q: Can I use the logarithmic base property to solve exponential equations with complex numbers?

A: Yes, you can use the logarithmic base property to solve exponential equations with complex numbers. However, you need to be careful when taking the logarithm of both sides of the equation, as this will give you a complex number.

Q: What are some common errors to avoid when using the logarithmic base property?

A: Some common errors to avoid when using the logarithmic base property include:

Not using the correct base: This will give you an incorrect solution.
Not taking the logarithm of both sides of the equation: This will give you an incorrect solution.
Not using the change of base formula: This will give you an incorrect solution.

Conclusion

In conclusion, the logarithmic base property is a powerful tool for solving exponential equations. By using the logarithmic base property, you can take the logarithm of both sides of the equation and then use the change of base formula to evaluate the logarithm. This will give you the value of the exponent, which is the solution to the equation.