Solving Exponential Equations Using The Like Bases Property $3^{n+10}=3^{7n+7}$

Jul 14, 2025 by ADMIN 80 views

Solve Exponential Equations Using the Like Bases Property

Solving exponential equations might seem daunting at first, but with the right techniques, they can become quite manageable. One powerful tool in our arsenal is the like bases property. This property allows us to simplify equations where exponential expressions with the same base are set equal to each other. In this article, we will delve into the like bases property and demonstrate its application by solving the equation $3^{n+10} = 3^{7n+7}$ . We'll break down each step to ensure a clear understanding of the process, and by the end, you'll be well-equipped to tackle similar problems. So, let's embark on this mathematical journey together!

Understanding the Like Bases Property

The like bases property is a fundamental concept in algebra that enables us to solve exponential equations efficiently. This property states that if two exponential expressions with the same base are equal, then their exponents must also be equal. Mathematically, this can be expressed as follows:

If $b^x = b^y$ , then $x = y$ , where b is a positive number not equal to 1.

This property is derived from the fact that exponential functions are one-to-one. In simpler terms, for a given base, each exponent corresponds to a unique value. Therefore, if two exponential expressions with the same base yield the same value, their exponents must be identical.

The importance of the like bases property cannot be overstated when it comes to solving exponential equations. It transforms complex exponential equations into simpler algebraic equations that are much easier to solve. By equating the exponents, we eliminate the exponential terms, allowing us to focus on solving for the variable using standard algebraic techniques.

To illustrate the power of this property, consider a simple example. Suppose we have the equation $2^x = 2^3$ . Applying the like bases property, we can directly conclude that x = 3. This demonstrates how the property allows us to bypass complex calculations and arrive at the solution swiftly.

In the context of more complex equations, the like bases property serves as a cornerstone for simplifying and solving them. It is particularly useful when dealing with exponential equations where the variable appears in the exponent. By recognizing and applying this property, we can transform seemingly intricate equations into manageable algebraic expressions, paving the way for finding the solution.

Applying the Like Bases Property to the Equation

Now that we have a solid grasp of the like bases property, let's apply it to solve the given equation: $3^{n+10} = 3^{7n+7}$ .

The first step in solving any exponential equation is to ensure that the bases are the same. In this case, we are fortunate because both sides of the equation already have the same base, which is 3. This allows us to directly apply the like bases property.

According to the like bases property, if $b^x = b^y$ , then $x = y$ . Applying this to our equation, we can equate the exponents:

$n + 10 = 7n + 7$

Now we have a simple linear equation in terms of 'n'. Our next step is to solve this equation for 'n'. To do this, we'll use standard algebraic techniques to isolate 'n' on one side of the equation.

First, let's subtract 'n' from both sides of the equation:

$10 = 6n + 7$

Next, we'll subtract 7 from both sides:

$3 = 6n$

Finally, we'll divide both sides by 6 to solve for 'n':

$n = rac{3}{6}$

Simplifying the fraction, we get:

$n = rac{1}{2}$

Therefore, the solution to the equation $3^{n+10} = 3^{7n+7}$ is $n = rac{1}{2}$ . This result demonstrates the effectiveness of the like bases property in simplifying exponential equations and finding their solutions.

Step-by-Step Solution: Solving for n

To further clarify the process, let's break down the steps involved in solving for 'n' in the equation $3^{n+10} = 3^{7n+7}$ .

Equate the exponents:

Since the bases are the same (both are 3), we can apply the like bases property and equate the exponents:

$n + 10 = 7n + 7$

This step is the crucial application of the like bases property, transforming the exponential equation into a linear equation.
Subtract 'n' from both sides:

To isolate the terms with 'n' on one side, we subtract 'n' from both sides of the equation:

$n + 10 - n = 7n + 7 - n$

This simplifies to:

$10 = 6n + 7$

This step helps to group the 'n' terms on one side of the equation, moving us closer to isolating 'n'.
Subtract 7 from both sides:

To further isolate the term with 'n', we subtract 7 from both sides of the equation:

$10 - 7 = 6n + 7 - 7$

This simplifies to:

$3 = 6n$

This step isolates the term with 'n' on one side of the equation, making it easier to solve for 'n'.
Divide both sides by 6:

To solve for 'n', we divide both sides of the equation by 6:

$rac{3}{6} = rac{6n}{6}$

This simplifies to:

$n = rac{1}{2}$

This step isolates 'n' and provides us with the solution to the equation.
Simplify the fraction:

The fraction $rac{3}{6}$ can be simplified to $rac{1}{2}$ .

Therefore, the final solution is:

$n = rac{1}{2}$

This step ensures that the answer is presented in its simplest form, adhering to mathematical conventions.

By following these steps, we have successfully solved for 'n' in the equation $3^{n+10} = 3^{7n+7}$ . This step-by-step breakdown demonstrates the methodical approach to solving exponential equations using the like bases property.

Expressing the Solution as an Integer or Reduced Fraction

In the previous sections, we successfully solved the equation $3^{n+10} = 3^{7n+7}$ and arrived at the solution $n = rac{1}{2}$ . Now, let's address the requirement of expressing the answer as an integer or a reduced fraction.

Our solution, $n = rac{1}{2}$ , is already in the form of a fraction. However, we need to ensure that it is a reduced fraction. A reduced fraction, also known as a simplified fraction, is a fraction where the numerator and denominator have no common factors other than 1. In other words, the fraction cannot be simplified any further.

In our case, the fraction $rac{1}{2}$ has a numerator of 1 and a denominator of 2. The only common factor between 1 and 2 is 1, which means that the fraction is already in its simplest form. Therefore, $rac{1}{2}$ is a reduced fraction.

The problem also asks us to express the answer as an integer if possible. An integer is a whole number (not a fraction). Since $rac{1}{2}$ is not a whole number, it cannot be expressed as an integer.

Therefore, the final answer, expressed as an integer or reduced fraction, is:

$n = rac{1}{2}$

This solution satisfies the requirement of being expressed as a reduced fraction. It is important to always check if a fraction can be simplified further to ensure that the answer is presented in its most concise form.

Practice Problems: Mastering the Technique

To solidify your understanding of the like bases property and its application in solving exponential equations, let's explore some practice problems. These problems will provide you with an opportunity to apply the techniques we've discussed and build your confidence in solving similar equations.

Practice Problem 1:

Solve for x: $5^{2x-1} = 5^{x+3}$

Solution:

Since the bases are the same (both are 5), we can equate the exponents:

$2x - 1 = x + 3$

Now, let's solve for x:

Subtract x from both sides:

$2x - 1 - x = x + 3 - x$

$x - 1 = 3$

Add 1 to both sides:

$x - 1 + 1 = 3 + 1$

$x = 4$

Therefore, the solution is $x = 4$ .

Practice Problem 2:

Solve for y: $2^{3y+2} = 2^{5y-4}$

Solution:

Since the bases are the same (both are 2), we can equate the exponents:

$3y + 2 = 5y - 4$

Now, let's solve for y:

Subtract 3y from both sides:

$3y + 2 - 3y = 5y - 4 - 3y$

$2 = 2y - 4$

Add 4 to both sides:

$2 + 4 = 2y - 4 + 4$

$6 = 2y$

Divide both sides by 2:

$rac{6}{2} = rac{2y}{2}$

$y = 3$

Therefore, the solution is $y = 3$ .

Practice Problem 3:

Solve for z: $4^{z+1} = 4^{2z-3}$

Solution:

Since the bases are the same (both are 4), we can equate the exponents:

$z + 1 = 2z - 3$

Now, let's solve for z:

Subtract z from both sides:

$z + 1 - z = 2z - 3 - z$

$1 = z - 3$

Add 3 to both sides:

$1 + 3 = z - 3 + 3$

$4 = z$

Therefore, the solution is $z = 4$ .

These practice problems demonstrate the consistent application of the like bases property in solving exponential equations. By equating the exponents and solving the resulting algebraic equation, we can efficiently find the solutions. Remember to always check your answers by substituting them back into the original equation to ensure accuracy.

Conclusion

In this comprehensive guide, we've explored the like bases property and its application in solving exponential equations. We've seen how this property allows us to simplify equations with the same base by equating their exponents, transforming complex exponential problems into simpler algebraic ones. By following a step-by-step approach, we successfully solved the equation $3^{n+10} = 3^{7n+7}$ and found the solution $n = rac{1}{2}$ . We also addressed the requirement of expressing the answer as an integer or reduced fraction, ensuring that the solution was presented in its simplest form.

Furthermore, we worked through several practice problems to solidify your understanding of the technique. These problems provided valuable opportunities to apply the like bases property and build confidence in solving similar equations. By mastering this technique, you'll be well-equipped to tackle a wide range of exponential equations.

Remember, the key to success in mathematics lies in consistent practice and a thorough understanding of fundamental concepts. The like bases property is a powerful tool in your mathematical arsenal, and with continued practice, you'll become proficient in its application. So, keep practicing, keep exploring, and keep pushing your mathematical boundaries!