Solve For $x$ To The Nearest Tenth.$16=980(0.5)^{\frac{x}{8}}$

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic and exponential functions. In this article, we will focus on solving a specific exponential equation, $16=980(0.5)^{\frac{x}{8}}$, to the nearest tenth. We will break down the solution into manageable steps, making it easier for readers to understand and apply the concepts.

Understanding Exponential Functions

Before we dive into solving the equation, let's take a moment to understand exponential functions. An exponential function is a function that has the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable. The graph of an exponential function is a curve that increases or decreases rapidly as $x$ increases or decreases.

The Given Equation

The given equation is $16=980(0.5)^{\frac{x}{8}}$. To solve for $x$, we need to isolate the variable $x$ on one side of the equation. We can start by dividing both sides of the equation by $980$, which gives us:

16980=(0.5)x8\frac{16}{980} = (0.5)^{\frac{x}{8}}

Simplifying the Equation

Next, we can simplify the left-hand side of the equation by dividing $16$ by $980$. This gives us:

16980=161.25\frac{16}{980} = \frac{1}{61.25}

Now, we can rewrite the equation as:

161.25=(0.5)x8\frac{1}{61.25} = (0.5)^{\frac{x}{8}}

Using Logarithms to Solve the Equation

To solve for $x$, we can use logarithms to get rid of the exponent. We can take the logarithm of both sides of the equation, which gives us:

log⁡(161.25)=log⁡((0.5)x8)\log\left(\frac{1}{61.25}\right) = \log\left((0.5)^{\frac{x}{8}}\right)

Using the property of logarithms that $\log(a^b) = b\log(a)$, we can rewrite the equation as:

log⁡(161.25)=x8log⁡(0.5)\log\left(\frac{1}{61.25}\right) = \frac{x}{8}\log(0.5)

Solving for x

Now, we can solve for $x$ by multiplying both sides of the equation by $8$, which gives us:

8log⁡(161.25)=xlog⁡(0.5)8\log\left(\frac{1}{61.25}\right) = x\log(0.5)

Next, we can divide both sides of the equation by $\log(0.5)$, which gives us:

x=8log⁡(161.25)log⁡(0.5)x = \frac{8\log\left(\frac{1}{61.25}\right)}{\log(0.5)}

Evaluating the Expression

To evaluate the expression, we can use a calculator to find the values of the logarithms. Plugging in the values, we get:

x=8log⁡(161.25)log⁡(0.5)≈8(−1.19)−0.70≈13.4x = \frac{8\log\left(\frac{1}{61.25}\right)}{\log(0.5)} \approx \frac{8(-1.19)}{-0.70} \approx 13.4

Conclusion

In this article, we solved the exponential equation $16=980(0.5)^{\frac{x}{8}}$ to the nearest tenth. We used logarithms to get rid of the exponent and solve for $x$. The final answer is $x \approx 13.4$.

Tips and Tricks

  • When solving exponential equations, it's essential to use logarithms to get rid of the exponent.
  • Make sure to simplify the equation before using logarithms.
  • Use a calculator to evaluate the expression and find the value of $x$.

Practice Problems

  • Solve the exponential equation $32=512(2)^{\frac{x}{3}}$ to the nearest tenth.
  • Solve the exponential equation $81=729(3)^{\frac{x}{4}}$ to the nearest tenth.

References

  • [1] "Exponential Functions" by Math Open Reference
  • [2] "Logarithmic Functions" by Math Open Reference

Glossary

  • Exponential function: A function that has the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable.
  • Logarithmic function: A function that has the form $f(x) = \log(a^x)$, where $a$ is a positive constant and $x$ is the variable.
  • Exponent: A number that is raised to a power, such as $2^3$.
  • Logarithm: The inverse of an exponential function, such as $\log(2^3) = 3$.

Introduction

In our previous article, we solved the exponential equation $16=980(0.5)^{\frac{x}{8}}$ to the nearest tenth. We used logarithms to get rid of the exponent and solve for $x$. In this article, we will answer some frequently asked questions about solving exponential equations.

Q: What is an exponential equation?

A: An exponential equation is an equation that contains an exponential expression, such as $a^x$, where $a$ is a positive constant and $x$ is the variable.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use logarithms to get rid of the exponent. You can take the logarithm of both sides of the equation, which gives you:

log⁡(ax)=xlog⁡(a)\log(a^x) = x\log(a)

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

A: An exponential function is a function that has the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable. A logarithmic function is a function that has the form $f(x) = \log(a^x)$, where $a$ is a positive constant and $x$ is the variable.

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

A: To use logarithms to solve an exponential equation, you can take the logarithm of both sides of the equation. This gives you:

log⁡(ax)=xlog⁡(a)\log(a^x) = x\log(a)

You can then solve for $x$ by dividing both sides of the equation by $\log(a)$.

Q: What is the base of a logarithm?

A: The base of a logarithm is the number that is raised to a power in the exponential expression. For example, in the equation $\log(2^x)$, the base is $2$.

Q: How do I choose the base of a logarithm?

A: You can choose any base for a logarithm, but the most common bases are $10$ and $e$. The base $10$ is often used in scientific notation, while the base $e$ is often used in calculus.

Q: What is the difference between a natural logarithm and a common logarithm?

A: A natural logarithm is a logarithm with a base of $e$, while a common logarithm is a logarithm with a base of $10$.

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 press the "solve" button. The calculator will then give you the value of $x$.

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

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

  • Not using logarithms to get rid of the exponent
  • Not simplifying the equation before using logarithms
  • Not checking the domain of the logarithmic function
  • Not using a calculator to check the answer

Conclusion

In this article, we answered some frequently asked questions about solving exponential equations. We covered topics such as the definition of an exponential equation, how to solve an exponential equation using logarithms, and common mistakes to avoid. We hope that this article has been helpful in answering your questions about solving exponential equations.

Practice Problems

  • Solve the exponential equation $32=512(2)^{\frac{x}{3}}$ to the nearest tenth.
  • Solve the exponential equation $81=729(3)^{\frac{x}{4}}$ to the nearest tenth.

References

  • [1] "Exponential Functions" by Math Open Reference
  • [2] "Logarithmic Functions" by Math Open Reference

Glossary

  • Exponential function: A function that has the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable.
  • Logarithmic function: A function that has the form $f(x) = \log(a^x)$, where $a$ is a positive constant and $x$ is the variable.
  • Exponent: A number that is raised to a power, such as $2^3$.
  • Logarithm: The inverse of an exponential function, such as $\log(2^3) = 3$.
  • Base: The number that is raised to a power in the exponential expression.
  • Natural logarithm: A logarithm with a base of $e$.
  • Common logarithm: A logarithm with a base of $10$.

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