Forgetting to drop the logarithm

Oct 13, 2025 by miodeya.com 7 views

Introduction

In mathematics, logarithmic functions play a crucial role in various mathematical operations and applications. The logarithmic function is the inverse of the exponential function, and it is used to solve equations involving exponential expressions. In this article, we will explore the concept of logarithmic functions and how to find a value for $x$ that satisfies the equation $f(x) = g(x)$ , where $f(x) = \log_4(2x - 7)$ and $g(x) = \log_4 3$ .

Understanding Logarithmic Functions

A logarithmic function is a mathematical function that is the inverse of an exponential function. It is used to solve equations involving exponential expressions. The logarithmic function is defined as:

\log_b a = c \iff b^c = a

where $b$ is the base of the logarithm, $a$ is the argument of the logarithm, and $c$ is the result of the logarithm.

In this case, we have two logarithmic functions:

f(x) = \log_4(2x - 7)

g(x) = \log_4 3

Setting Up the Equation

To find a value for $x$ that satisfies the equation $f(x) = g(x)$ , we need to set up the equation and solve for $x$ . We can do this by equating the two logarithmic functions:

\log_4(2x - 7) = \log_4 3

Using the One-to-One Property of Logarithms

Since the logarithmic function is one-to-one, we can drop the logarithm and equate the arguments:

2x - 7 = 3

Solving for $x$

Now, we can solve for $x$ by adding 7 to both sides of the equation:

2x = 10

Next, we can divide both sides of the equation by 2:

x = 5

Conclusion

In this article, we have explored the concept of logarithmic functions and how to find a value for $x$ that satisfies the equation $f(x) = g(x)$ , where $f(x) = \log_4(2x - 7)$ and $g(x) = \log_4 3$ . We have used the one-to-one property of logarithms to drop the logarithm and equate the arguments, and then solved for $x$ by adding 7 to both sides of the equation and dividing both sides by 2. The value of $x$ that satisfies the equation is $x = 5$ .

Example Use Cases

Logarithmic functions have many practical applications in various fields, including:

Finance: Logarithmic functions are used to calculate interest rates and returns on investments.
Science: Logarithmic functions are used to model population growth and decay, and to calculate the pH of a solution.
Engineering: Logarithmic functions are used to design and optimize systems, such as electronic circuits and mechanical systems.

Tips and Tricks

Use the one-to-one property of logarithms: When solving equations involving logarithmic functions, use the one-to-one property to drop the logarithm and equate the arguments.
Check your units: When working with logarithmic functions, make sure to check your units to ensure that they are consistent.
Use a calculator: When working with logarithmic functions, use a calculator to check your answers and to ensure that you are getting the correct results.

Common Mistakes

Forgetting to drop the logarithm: When solving equations involving logarithmic functions, make sure to drop the logarithm using the one-to-one property.
Not checking units: When working with logarithmic functions, make sure to check your units to ensure that they are consistent.
Not using a calculator: When working with logarithmic functions, use a calculator to check your answers and to ensure that you are getting the correct results.

Conclusion

In conclusion, logarithmic functions are a powerful tool for solving equations involving exponential expressions. By using the one-to-one property of logarithms and checking our units, we can solve equations involving logarithmic functions and find values for $x$ that satisfy the equation. With practice and experience, we can become proficient in working with logarithmic functions and apply them to real-world problems.

Introduction

In our previous article, we explored the concept of logarithmic functions and how to find a value for $x$ that satisfies the equation $f(x) = g(x)$ , where $f(x) = \log_4(2x - 7)$ and $g(x) = \log_4 3$ . In this article, we will answer some frequently asked questions about logarithmic functions and provide additional examples and tips for working with logarithmic functions.

Q&A

Q: What is the one-to-one property of logarithms?

A: The one-to-one property of logarithms states that if $f(x) = g(x)$ , then $x = y$ , where $f(x)$ and $g(x)$ are logarithmic functions with the same base.

Q: How do I drop the logarithm using the one-to-one property?

A: To drop the logarithm using the one-to-one property, you need to equate the arguments of the two logarithmic functions. For example, if $f(x) = \log_4(2x - 7)$ and $g(x) = \log_4 3$ , you can drop the logarithm by equating the arguments:

2x - 7 = 3

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

A: A logarithmic function is the inverse of an exponential function. While an exponential function raises a number to a power, a logarithmic function finds the power to which a number must be raised to produce a given value.

Q: How do I solve an equation involving logarithmic functions?

A: To solve an equation involving logarithmic functions, you need to use the one-to-one property to drop the logarithm and equate the arguments. Then, you can solve for the variable using algebraic methods.

Q: What are some common mistakes to avoid when working with logarithmic functions?

A: Some common mistakes to avoid when working with logarithmic functions include:

Forgetting to drop the logarithm using the one-to-one property
Not checking units to ensure that they are consistent
Not using a calculator to check answers and ensure that they are correct

Q: What are some real-world applications of logarithmic functions?

A: Logarithmic functions have many practical applications in various fields, including:

Finance: Logarithmic functions are used to calculate interest rates and returns on investments.
Science: Logarithmic functions are used to model population growth and decay, and to calculate the pH of a solution.
Engineering: Logarithmic functions are used to design and optimize systems, such as electronic circuits and mechanical systems.

Additional Examples

Example 1: Solving an Equation Involving Logarithmic Functions

Solve the equation:

\log_4(2x - 7) = \log_4 5

Using the one-to-one property, we can drop the logarithm and equate the arguments:

2x - 7 = 5

Solving for $x$ , we get:

x = 6

Example 2: Finding the Value of a Logarithmic Function

Find the value of:

\log_4 24

Using a calculator, we can find the value of the logarithmic function:

\log_4 24 \approx 3.17

Tips and Tricks

Use the one-to-one property of logarithms to drop the logarithm and equate the arguments.
Check your units to ensure that they are consistent.
Use a calculator to check answers and ensure that they are correct.
Practice working with logarithmic functions to become proficient.