Solve The Equation For { T $} : : : { 10^{t+1} = 50 \}

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving the equation $10^{t+1} = 50$. This equation involves an exponential function with a base of 10, and our goal is to isolate the variable $t$.

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 of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable. The base $a$ determines the rate at which the function grows or decays. In our equation, the base is 10, which means that the function will grow rapidly as $t$ increases.

Solving the Equation

Now that we have a basic understanding of exponential functions, let's focus on solving the equation $10^{t+1} = 50$. To solve this equation, we will use the following steps:

Step 1: Isolate the Exponential Term

The first step in solving the equation is to isolate the exponential term. We can do this by dividing both sides of the equation by 10:

$$\frac{10^{t+1}}{10} = \frac{50}{10}$$

This simplifies to:

$$10^t = 5$$

Step 2: Use Logarithms to Solve for $t$

Now that we have isolated the exponential term, we can use logarithms to solve for $t$. We will use the logarithm base 10, which is denoted by $\log_{10}$. Taking the logarithm base 10 of both sides of the equation, we get:

$$\log_{10}(10^t) = \log_{10}(5)$$

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

$$t = \log_{10}(5)$$

Step 3: Evaluate the Logarithm

Now that we have solved for $t$, we need to evaluate the logarithm. Using a calculator or a logarithm table, we can find that:

$$\log_{10}(5) \approx 0.69897$$

Therefore, the value of $t$ is approximately 0.69897.

Conclusion

Solving the equation $10^{t+1} = 50$ involves isolating the exponential term, using logarithms to solve for $t$, and evaluating the logarithm. By following these steps, we can find the value of $t$ that satisfies the equation. This equation is a simple example of an exponential equation, and solving it requires a basic understanding of exponential functions and logarithms.

Real-World Applications

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 proportional to the current population.
• Financial modeling: Exponential equations can be used to model financial growth, where the value of an investment grows at a rate proportional to the current value.
• Science and engineering: Exponential equations can be used to model many scientific and engineering phenomena, including chemical reactions, electrical circuits, and population dynamics.

Tips and Tricks

Here are some tips and tricks for solving exponential equations:

• Use logarithms: Logarithms are a powerful tool for solving exponential equations. By taking the logarithm of both sides of the equation, we can isolate the variable and solve for it.
• Check your units: When solving exponential equations, it's essential to check your units. Make sure that the units on both sides of the equation are consistent.
• Use a calculator: A calculator can be a valuable tool for solving exponential equations. Use it to evaluate logarithms and exponential functions.

Common Mistakes

Here are some common mistakes to avoid when solving exponential equations:

• Not isolating the exponential term: Make sure to isolate the exponential term before using logarithms to solve for the variable.
• Not checking units: Check your units to ensure that they are consistent on both sides of the equation.
• Not using logarithms: Logarithms are a powerful tool for solving exponential equations. Don't be afraid to use them!

Conclusion

$$$$$$

Introduction

In our previous article, we discussed how to solve the equation $10^{t+1} = 50$. We used logarithms to isolate the variable $t$ and found that $t \approx 0.69897$. In this article, we will answer some common questions about solving exponential equations.

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 constant and $x$ is the variable.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the exponential term and then use logarithms to solve for the variable. Here are the steps:

1. Isolate the exponential term.
2. Take the logarithm of both sides of the equation.
3. Use the property of logarithms that $\log_a(a^x) = x$ to simplify the equation.
4. Solve for the variable.

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 a function of the form $f(x) = \log_a(x)$, where $a$ is a positive constant and $x$ is the variable. An exponential equation, on the other hand, is an equation that involves an exponential function, which is a function of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable.

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

A: Yes, you can use a calculator to solve an exponential equation. In fact, a calculator can be a valuable tool for solving exponential equations. You can use it to evaluate logarithms and exponential functions.

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

A: Here are some common mistakes to avoid when solving exponential equations:

• Not isolating the exponential term.
• Not checking units.
• Not using logarithms.
• Not simplifying the equation using the property of logarithms that $\log_a(a^x) = x$.

Q: Can I use a graphing calculator to solve an exponential equation?

A: Yes, you can use a graphing calculator to solve an exponential equation. In fact, a graphing calculator can be a powerful tool for solving exponential equations. You can use it to graph the equation and find the solution.

Q: How do I check my work when solving an exponential equation?

A: Here are some steps you can follow to check your work when solving an exponential equation:

1. Plug the solution back into the original equation.
2. Simplify the equation.
3. Check that the solution satisfies the equation.

Q: Can I use a computer algebra system (CAS) to solve an exponential equation?

A: Yes, you can use a computer algebra system (CAS) to solve an exponential equation. In fact, a CAS can be a powerful tool for solving exponential equations. You can use it to solve the equation and find the solution.

Conclusion

Solving exponential equations can be a challenging task, but with the right tools and techniques, it can be done. In this article, we have answered some common questions about solving exponential equations and provided some tips and tricks for solving them. We hope that this article has been helpful in your studies.

Real-World Applications

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 proportional to the current population.
• Financial modeling: Exponential equations can be used to model financial growth, where the value of an investment grows at a rate proportional to the current value.
• Science and engineering: Exponential equations can be used to model many scientific and engineering phenomena, including chemical reactions, electrical circuits, and population dynamics.

Tips and Tricks

Here are some tips and tricks for solving exponential equations:

• Use logarithms: Logarithms are a powerful tool for solving exponential equations. By taking the logarithm of both sides of the equation, we can isolate the variable and solve for it.
• Check your units: When solving exponential equations, it's essential to check your units. Make sure that the units on both sides of the equation are consistent.
• Use a calculator: A calculator can be a valuable tool for solving exponential equations. Use it to evaluate logarithms and exponential functions.

Common Mistakes

Here are some common mistakes to avoid when solving exponential equations:

• Not isolating the exponential term: Make sure to isolate the exponential term before using logarithms to solve for the variable.
• Not checking units: Check your units to ensure that they are consistent on both sides of the equation.
• Not using logarithms: Logarithms are a powerful tool for solving exponential equations. Don't be afraid to use them!

Conclusion

Solving exponential equations can be a challenging task, but with the right tools and techniques, it can be done. In this article, we have answered some common questions about solving exponential equations and provided some tips and tricks for solving them. We hope that this article has been helpful in your studies.