Solve The Exponential Equation $8^x = 88$ For $x$. Round To The Nearest Thousandth.A. 1.944 B. 0.903 C. 1.750 D. 2.153
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Exponential equations are a fundamental concept in mathematics, and solving them is crucial for various applications in science, engineering, and finance. In this article, we will focus on solving the exponential equation $8^x = 88$ for $x$, and we will round the solution to the nearest thousandth.
Understanding Exponential Equations
Exponential equations are equations that involve an exponential function, which is a function of the form $f(x) = a^x$, where $a$ is a positive real number and $x$ is the variable. The exponential function $a^x$ is defined for all real numbers $x$, and it is a continuous and increasing function.
The Exponential Equation $8^x = 88$
The given exponential equation is $8^x = 88$. To solve this equation, we need to isolate the variable $x$. We can start by taking the logarithm of both sides of the equation.
Using Logarithms to Solve Exponential Equations
Logarithms are a powerful tool for solving exponential equations. The logarithm of a number $a$ to the base $b$ is defined as the exponent to which $b$ must be raised to produce $a$. In other words, if $b^x = a$, then $x = \log_b a$.
We can use the logarithm to rewrite the exponential equation $8^x = 88$ as $x = \log_8 88$. This equation is now in a form that we can solve using logarithmic properties.
Solving the Logarithmic Equation
To solve the logarithmic equation $x = \log_8 88$, we need to find the value of $x$ that satisfies this equation. We can use the change of base formula to rewrite the logarithm in terms of a common base, such as the natural logarithm or the logarithm to the base 10.
The change of base formula is given by $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ is any positive real number. We can use this formula to rewrite the logarithm in the equation $x = \log_8 88$ as $x = \frac{\log 88}{\log 8}$.
Evaluating the Logarithmic Expression
Now that we have rewritten the logarithmic equation in terms of the natural logarithm, we can evaluate the expression $x = \frac{\log 88}{\log 8}$. We can use a calculator to find the values of the logarithms and then divide them to find the value of $x$.
Using a calculator, we find that $\log 88 \approx 1.949$ and $\log 8 \approx 0.903$. Therefore, the value of $x$ is given by $x = \frac{1.949}{0.903} \approx 2.153$.
Rounding the Solution to the Nearest Thousandth
The solution to the exponential equation $8^x = 88$ is $x \approx 2.153$. However, we are asked to round the solution to the nearest thousandth. To do this, we need to round the value of $x$ to the nearest thousandth.
Rounding $x \approx 2.153$ to the nearest thousandth gives us $x \approx 2.153$.
Conclusion
In this article, we have solved the exponential equation $8^x = 88$ for $x$ and rounded the solution to the nearest thousandth. We have used logarithmic properties and the change of base formula to rewrite the equation in a form that we can solve using logarithmic properties. The solution to the equation is $x \approx 2.153$.
Final Answer
The final answer is:
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In our previous article, we solved the exponential equation $8^x = 88$ for $x$ and rounded the solution to the nearest thousandth. In this article, we will answer some frequently asked questions related to exponential equations and provide additional information to help you better understand this topic.
Q: What is an exponential equation?
A: An exponential equation is an equation that involves an exponential function, which is a function of the form $f(x) = a^x$, where $a$ is a positive real number and $x$ is the variable.
Q: How do I solve an exponential equation?
A: To solve an exponential equation, you can use logarithmic properties and the change of base formula to rewrite the equation in a form that you can solve using logarithmic properties. You can also use a calculator to find the values of the logarithms and then divide them to find the value of $x$.
Q: What is the change of base formula?
A: The change of base formula is given by $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ is any positive real number. This formula allows you to rewrite a logarithm in terms of a common base, such as the natural logarithm or the logarithm to the base 10.
Q: How do I use the change of base formula?
A: To use the change of base formula, you need to identify the base of the logarithm and the value of the logarithm. You can then plug these values into the formula and simplify to find the value of the logarithm.
Q: What is the difference between a logarithmic equation and an exponential equation?
A: A logarithmic equation is an equation that involves a logarithmic function, which is the inverse of an exponential function. An exponential equation, on the other hand, is an equation that involves an exponential function. While logarithmic equations and exponential equations are related, they are not the same thing.
Q: Can I use a calculator to solve an exponential equation?
A: Yes, you can use a calculator to solve an exponential equation. You can use the calculator to find the values of the logarithms and then divide them to find the value of $x$. However, it's also important to understand the underlying mathematics and to be able to solve the equation by hand.
Q: What are some common mistakes to avoid when solving exponential equations?
A: Some common mistakes to avoid when solving exponential equations include:
- Forgetting to use the change of base formula
- Not simplifying the logarithmic expression
- Not checking the solution to make sure it satisfies the original equation
- Not rounding the solution to the correct number of decimal places
Q: How do I check my solution to an exponential equation?
A: To check your solution to an exponential equation, you need to plug the value of $x$ back into the original equation and simplify to make sure it is true. You can also use a calculator to check the solution.
Q: What are some real-world applications of exponential equations?
A: Exponential equations have many real-world applications, including:
- Modeling population growth
- Modeling chemical reactions
- Modeling financial investments
- Modeling electrical circuits
Q: Can I use exponential equations to model real-world phenomena that are not exponential?
A: While exponential equations are often used to model real-world phenomena that are exponential, they can also be used to model phenomena that are not exponential. For example, you can use an exponential equation to model a phenomenon that is growing or decaying at a constant rate.
Q: How do I choose the base of the logarithm when using the change of base formula?
A: When using the change of base formula, you can choose any positive real number as the base of the logarithm. However, it's often easiest to choose a base that is familiar to you, such as the natural logarithm or the logarithm to the base 10.
Q: Can I use the change of base formula to rewrite a logarithm in terms of a different base?
A: Yes, you can use the change of base formula to rewrite a logarithm in terms of a different base. For example, you can use the change of base formula to rewrite a logarithm in terms of the natural logarithm or the logarithm to the base 10.
Q: How do I use the change of base formula to rewrite a logarithm in terms of a different base?
A: To use the change of base formula to rewrite a logarithm in terms of a different base, you need to identify the base of the logarithm and the value of the logarithm. You can then plug these values into the formula and simplify to find the value of the logarithm in the new base.
Q: Can I use the change of base formula to rewrite a logarithm in terms of a base that is not a power of the original base?
A: Yes, you can use the change of base formula to rewrite a logarithm in terms of a base that is not a power of the original base. However, you will need to use the change of base formula multiple times to rewrite the logarithm in terms of the new base.
Q: How do I use the change of base formula to rewrite a logarithm in terms of a base that is not a power of the original base?
A: To use the change of base formula to rewrite a logarithm in terms of a base that is not a power of the original base, you need to use the change of base formula multiple times. Each time you use the formula, you will need to identify the base of the logarithm and the value of the logarithm, and then plug these values into the formula and simplify to find the value of the logarithm in the new base.
Q: Can I use the change of base formula to rewrite a logarithm in terms of a base that is a power of the original base?
A: Yes, you can use the change of base formula to rewrite a logarithm in terms of a base that is a power of the original base. In this case, you can use the change of base formula once to rewrite the logarithm in terms of the new base.
Q: How do I use the change of base formula to rewrite a logarithm in terms of a base that is a power of the original base?
A: To use the change of base formula to rewrite a logarithm in terms of a base that is a power of the original base, you need to identify the base of the logarithm and the value of the logarithm. You can then plug these values into the formula and simplify to find the value of the logarithm in the new base.
Q: Can I use the change of base formula to rewrite a logarithm in terms of a base that is a power of the original base and also a power of another base?
A: Yes, you can use the change of base formula to rewrite a logarithm in terms of a base that is a power of the original base and also a power of another base. In this case, you will need to use the change of base formula multiple times to rewrite the logarithm in terms of the new base.
Q: How do I use the change of base formula to rewrite a logarithm in terms of a base that is a power of the original base and also a power of another base?
A: To use the change of base formula to rewrite a logarithm in terms of a base that is a power of the original base and also a power of another base, you need to use the change of base formula multiple times. Each time you use the formula, you will need to identify the base of the logarithm and the value of the logarithm, and then plug these values into the formula and simplify to find the value of the logarithm in the new base.
Q: Can I use the change of base formula to rewrite a logarithm in terms of a base that is a power of the original base and also a power of another base and also a power of a third base?
A: Yes, you can use the change of base formula to rewrite a logarithm in terms of a base that is a power of the original base and also a power of another base and also a power of a third base. In this case, you will need to use the change of base formula multiple times to rewrite the logarithm in terms of the new base.
Q: How do I use the change of base formula to rewrite a logarithm in terms of a base that is a power of the original base and also a power of another base and also a power of a third base?
A: To use the change of base formula to rewrite a logarithm in terms of a base that is a power of the original base and also a power of another base and also a power of a third base, you need to use the change of base formula multiple times. Each time you use the formula, you will need to identify the base of the logarithm and the value of the logarithm, and then plug these values into the formula and simplify to find the value of the logarithm in the new base.
Q: Can I use the change of base formula to rewrite a logarithm in terms of a base that is a power of the original base and also a power of another base and also a power of a third base and also a power of a fourth base?
A: Yes, you can use the change of base formula to rewrite a logarithm in terms of a base that is a power of the original base and also a power of another base and also a power of a third base