Isolate the exponential term

Oct 13, 2025 by wbrum.org 64 views

=====================================================

Introduction

In this article, we will explore the process of solving an exponential equation algebraically and then verify the solution graphically. The given equation is $3300 = 300 \left(10^x\right)$ , and we need to find the value of $x$ that satisfies this equation.

Algebraic Solution

To solve the equation algebraically, we can start by isolating the exponential term on one side of the equation. We can do this by dividing both sides of the equation by 300.

\frac{3300}{300} = \frac{300 \left(10^x\right)}{300}

This simplifies to:

11 = 10^x

Next, we can take the logarithm of both sides of the equation to solve for $x$ . We can use the logarithm base 10, which is denoted by $\log_{10}$ .

\log_{10} 11 = \log_{10} 10^x

Using the property of logarithms that states $\log_b b^x = x$ , we can rewrite the right-hand side of the equation as:

\log_{10} 11 = x

Therefore, the value of $x$ that satisfies the equation is:

x = \log_{10} 11

Graphical Solution

To verify the solution graphically, we can plot the graph of the equation $y = 300 \left(10^x\right)$ and find the point where the graph intersects the horizontal line $y = 3300$ .

We can use a graphing calculator or a computer algebra system to plot the graph. The graph of the equation is a curve that increases rapidly as $x$ increases.

To find the point of intersection, we can set the equation equal to 3300 and solve for $x$ .

300 \left(10^x\right) = 3300

Dividing both sides of the equation by 300, we get:

10^x = \frac{3300}{300}

10^x = 11

Taking the logarithm base 10 of both sides, we get:

x = \log_{10} 11

This is the same solution we obtained algebraically.

Conclusion

In this article, we solved the equation $3300 = 300 \left(10^x\right)$ algebraically and verified the solution graphically. We found that the value of $x$ that satisfies the equation is $x = \log_{10} 11$ . This solution was verified graphically by plotting the graph of the equation and finding the point where the graph intersects the horizontal line $y = 3300$ .

Calculating the Value of $x$

To calculate the value of $x$ , we can use a calculator or a computer algebra system to evaluate the expression $\log_{10} 11$ .

Using a calculator, we get:

x \approx 1.0413927

Rounding to three decimal places, we get:

x \approx 1.041

Therefore, the value of $x$ that satisfies the equation is approximately $1.041$ .

Applications of Exponential Equations

Exponential equations have many applications in real-world problems. Some examples include:

Population growth: Exponential equations can be used to model the growth of populations over time.
Financial planning: Exponential equations can be used to calculate the future value of investments.
Chemical reactions: Exponential equations can be used to model the rate of chemical reactions.

Summary

References

Logarithm: A logarithm is the inverse operation of exponentiation. It is denoted by $\log_b a = x$ and is defined as the exponent to which the base $b$ must be raised to produce the number $a$ .
Exponential function: An exponential function is a function of the form $f(x) = a^x$ , where $a$ is a positive constant. It is used to model exponential growth or decay.
Graphing calculator: A graphing calculator is a calculator that can be used to plot the graph of a function. It is a useful tool for verifying solutions to equations graphically.

Introduction

In this article, we will answer some frequently asked questions about exponential equations. Exponential equations are a type of equation that involves an exponential function, which is a function of the form $f(x) = a^x$ , where $a$ is a positive constant.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential function. It is a type of equation that can be written in the form $a^x = b$ , where $a$ and $b$ are positive constants.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use the following steps:

Isolate the exponential term: Move all terms except the exponential term to the other side of the equation.
Take the logarithm of both sides: Take the logarithm of both sides of the equation to eliminate the exponential term.
Solve for x: Solve for $x$ using the properties of logarithms.

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

A: An exponential equation is an equation that involves an exponential function, while a logarithmic equation is an equation that involves a logarithmic function. Exponential equations are used to model exponential growth or decay, while logarithmic equations are used to model the inverse of exponential growth or decay.

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

A: Yes, you can use a calculator to solve an exponential equation. Many calculators have a built-in logarithm function that can be used to solve exponential equations.

Q: What are some common applications of exponential equations?

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

Population growth: Exponential equations can be used to model the growth of populations over time.
Financial planning: Exponential equations can be used to calculate the future value of investments.
Chemical reactions: Exponential equations can be used to model the rate of chemical reactions.

Q: How do I graph an exponential equation?

A: To graph an exponential equation, you can use a graphing calculator or a computer algebra system. You can also use a table of values to plot the graph of the equation.

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

A: An exponential function is a function of the form $f(x) = a^x$ , where $a$ is a positive constant. A power function is a function of the form $f(x) = x^n$ , where $n$ is a positive integer. Exponential functions grow or decay rapidly, while power functions grow or decay at a slower rate.

Q: Can I use a graphing calculator to verify the solution to an exponential equation?

A: Yes, you can use a graphing calculator to verify the solution to an exponential equation. You can plot the graph of the equation and find the point where the graph intersects the horizontal line $y = b$ .

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

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

Not isolating the exponential term: Make sure to isolate the exponential term on one side of the equation.
Not taking the logarithm of both sides: Make sure to take the logarithm of both sides of the equation to eliminate the exponential term.
Not using the correct properties of logarithms: Make sure to use the correct properties of logarithms to solve for $x$ .

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

A: To check your solution to an exponential equation, you can plug the value of $x$ back into the original equation and simplify. If the equation is true, then your solution is correct.

Q: What are some resources for learning more about exponential equations?

A: Some resources for learning more about exponential equations include:

Mathematics textbooks: There are many mathematics textbooks that cover exponential equations in detail. Some examples include "Calculus" by Michael Spivak and "Algebra" by Michael Artin.
Online resources: There are many online resources that provide tutorials and examples on exponential equations. Some examples include Khan Academy and MIT OpenCourseWare.
Graphing calculators: Many graphing calculators have built-in functions for solving exponential equations. Some examples include the TI-83 and the TI-84.