Select The Correct Answer.What Is The Approximate Solution To This Equation? 15 ( 3 ) 2 X = 90 15(3)^{2x} = 90 15 ( 3 ) 2 X = 90 A. X = 2 Log ⁡ 6 Log ⁡ 3 X = \frac{2 \log 6}{\log 3} X = L O G 3 2 L O G 6 ​ B. X = Log ⁡ 6 2 Log ⁡ 3 X = \frac{\log 6}{2 \log 3} X = 2 L O G 3 L O G 6 ​ C. X = 2 Log ⁡ 3 Log ⁡ 6 X = \frac{2 \log 3}{\log 6} X = L O G 6 2 L O G 3 ​ D. $x = \frac{\log 3}{2

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of logarithmic functions. In this article, we will explore how to solve exponential equations, with a focus on the given equation: $15(3)^{2x} = 90$. We will break down the solution step by step, using logarithmic properties to isolate the variable $x$.

Understanding Exponential Equations

Exponential equations involve a base raised to a power, which is then set equal to a constant. In this case, the equation is $15(3)^{2x} = 90$. To solve for $x$, we need to isolate the variable using logarithmic properties.

Step 1: Simplify the Equation

The first step is to simplify the equation by dividing both sides by 15. This gives us:

$$3^{2x} = \frac{90}{15}$$

Simplifying the right-hand side, we get:

$$3^{2x} = 6$$

Step 2: Use Logarithmic Properties

To solve for $x$, we can use logarithmic properties to isolate the variable. We can start by taking the logarithm of both sides of the equation. Using the property $\log a^b = b \log a$, we can rewrite the equation as:

$$2x \log 3 = \log 6$$

Step 3: Isolate the Variable

Now that we have the equation in terms of logarithms, we can isolate the variable $x$. We can do this by dividing both sides of the equation by $2 \log 3$. This gives us:

$$x = \frac{\log 6}{2 \log 3}$$

Conclusion

In this article, we have solved the exponential equation $15(3)^{2x} = 90$ using logarithmic properties. We have broken down the solution step by step, simplifying the equation and isolating the variable $x$. The final solution is:

$$x = \frac{\log 6}{2 \log 3}$$

This solution is in the form of a logarithmic expression, which is a common way to represent the solution to exponential equations.

Discussion

The solution to this equation is a classic example of how logarithmic properties can be used to solve exponential equations. By using the property $\log a^b = b \log a$, we can rewrite the equation in terms of logarithms, making it easier to isolate the variable.

Comparison of Solutions

Let's compare the solution we obtained with the options given:

• A. $x = \frac{2 \log 6}{\log 3}$
• B. $x = \frac{\log 6}{2 \log 3}$
• C. $x = \frac{2 \log 3}{\log 6}$
• D. $x = \frac{\log 3}{2 \log 6}$

The correct solution is option B: $x = \frac{\log 6}{2 \log 3}$.

Final Thoughts

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a base raised to a power, which is then set equal to a constant. For example, the equation $2^x = 8$ is an exponential equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use logarithmic properties to isolate the variable. You can start by taking the logarithm of both sides of the equation, and then use the property $\log a^b = b \log a$ to rewrite the equation in terms of logarithms.

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 expression. For example, the equation $\log x = 2$ is a logarithmic equation, while the equation $2^x = 8$ is an exponential equation.

Q: How do I use logarithmic properties to solve an exponential equation?

A: To use logarithmic properties to solve an exponential equation, you can start by taking the logarithm of both sides of the equation. Then, you can use the property $\log a^b = b \log a$ to rewrite the equation in terms of logarithms. Finally, you can isolate the variable by dividing both sides of the equation by the coefficient of the logarithm.

Q: What is the most common mistake people make when solving exponential equations?

A: The most common mistake people make when solving exponential equations is to forget to use logarithmic properties. This can lead to incorrect solutions or solutions that are not in the correct form.

Q: How do I check my solution to an exponential equation?

A: To check your solution to an exponential equation, you can plug the solution back into the original equation and see if it is true. If the solution is correct, then the equation will be true. If the solution is incorrect, then the equation will not be true.

Q: What are some common exponential equations that I should know how to solve?

A: Some common exponential equations that you should know how to solve include:

• $2^x = 8$
• $3^x = 27$
• $4^x = 64$
• $5^x = 125$

Q: How do I use a calculator to solve an exponential equation?

A: To use a calculator to solve an exponential equation, you can enter the equation into the calculator and use the logarithm function to solve for the variable. For example, to solve the equation $2^x = 8$, you can enter the equation into the calculator and use the logarithm function to get the solution $x = \log 8 / \log 2$.

Q: What are some real-world applications of exponential equations?

A: Exponential equations have many real-world applications, including:

• Population growth: Exponential equations can be used to model population growth, where the population grows at a rate that is proportional to the current population.
• Compound interest: Exponential equations can be used to calculate compound interest, where the interest is added to the principal at regular intervals.
• Radioactive decay: Exponential equations can be used to model radioactive decay, where the amount of radioactive material decreases at a rate that is proportional to the current amount.

Conclusion

In this article, we have answered some frequently asked questions about solving exponential equations. We have covered topics such as the definition of an exponential equation, how to solve an exponential equation using logarithmic properties, and how to check a solution to an exponential equation. We have also discussed some common exponential equations and real-world applications of exponential equations.