Identify the base of the logarithm

Oct 13, 2025 by miodeya.com 124 views

Introduction

In mathematics, transformations of functions are essential concepts that help us understand how functions change under various operations. The logarithmic function, specifically, is a fundamental function in mathematics that has numerous applications in various fields. In this article, we will explore the transformations of the logarithmic function $f(x) = \log_2 x$ and match each transformation with a feature of the transformed function.

Transformation 1: Reflection Across the x-axis

The first transformation we will consider is $g(x) = -2f(x)$ . To understand this transformation, let's break it down step by step.

The function $f(x) = \log_2 x$ is a logarithmic function with base 2.
The transformation $g(x) = -2f(x)$ $g (x) = - 2 f (x)$ involves two operations:
- Multiplication by $-2$ : This operation will stretch the function vertically by a factor of 2 and reflect it across the x-axis.
- The negative sign in front of the function will reflect the function across the x-axis.

The resulting function $g(x) = -2\log_2 x$ will have the following features:

Vertical Stretch: The function will be stretched vertically by a factor of 2, which means that the y-values will be twice as large as the original function.
Reflection Across the x-axis: The function will be reflected across the x-axis, which means that the y-values will be negated.

Transformation 2: Vertical Shift

The second transformation we will consider is $h(x) = f(x) + 2$ . This transformation involves adding 2 to the function $f(x) = \log_2 x$ .

The function $f(x) = \log_2 x$ is a logarithmic function with base 2.
The transformation $h(x) = f(x) + 2$ involves adding 2 to the function, which will shift the function vertically by 2 units.

The resulting function $h(x) = \log_2 x + 2$ will have the following features:

Vertical Shift: The function will be shifted vertically by 2 units, which means that the y-values will be 2 units larger than the original function.

Transformation 3: Horizontal Shift

The third transformation we will consider is $f(x-2)$ . This transformation involves replacing $x$ with $x-2$ in the function $f(x) = \log_2 x$ .

The function $f(x) = \log_2 x$ is a logarithmic function with base 2.
The transformation $f(x-2)$ involves replacing $x$ with $x-2$ , which will shift the function horizontally by 2 units to the right.

The resulting function $f(x-2) = \log_2 (x-2)$ will have the following features:

Horizontal Shift: The function will be shifted horizontally by 2 units to the right, which means that the x-values will be 2 units larger than the original function.

Conclusion

In conclusion, the transformations of the logarithmic function $f(x) = \log_2 x$ can be matched with the following features:

Reflection Across the x-axis: $g(x) = -2f(x)$
Vertical Shift: $h(x) = f(x) + 2$
Horizontal Shift: $f(x-2)$

These transformations are essential concepts in mathematics that help us understand how functions change under various operations. By understanding these transformations, we can analyze and solve problems involving logarithmic functions more effectively.

References

[1] Larson, R. (2019). Calculus. Cengage Learning.
[2] Rogawski, J. (2019). Calculus. W.H. Freeman and Company.
[3] Stewart, J. (2019). Calculus. Cengage Learning.

Glossary

Logarithmic Function: A function that is the inverse of an exponential function.
Transformation: A change in the function that involves a change in its graph.
Vertical Stretch: A transformation that stretches the function vertically.
Reflection Across the x-axis: A transformation that reflects the function across the x-axis.
Vertical Shift: A transformation that shifts the function vertically.
Horizontal Shift: A transformation that shifts the function horizontally.
Frequently Asked Questions (FAQs) About Transformations of Logarithmic Functions ====================================================================================

Q: What is a logarithmic function?

A: A logarithmic function is a function that is the inverse of an exponential function. It is a function that takes a positive real number as input and returns the exponent to which a fixed base must be raised to produce that number.

Q: What are the common transformations of logarithmic functions?

A: The common transformations of logarithmic functions include:

Reflection Across the x-axis: This transformation involves multiplying the function by a negative number, which reflects the function across the x-axis.
Vertical Stretch: This transformation involves multiplying the function by a positive number, which stretches the function vertically.
Vertical Shift: This transformation involves adding a constant to the function, which shifts the function vertically.
Horizontal Shift: This transformation involves replacing x with x-a, where a is a constant, which shifts the function horizontally.

Q: How do I apply the reflection across the x-axis transformation to a logarithmic function?

A: To apply the reflection across the x-axis transformation to a logarithmic function, you need to multiply the function by a negative number. For example, if you have the function f(x) = log2 x, you can apply the reflection across the x-axis transformation by multiplying it by -1, resulting in the function g(x) = -log2 x.

Q: How do I apply the vertical stretch transformation to a logarithmic function?

A: To apply the vertical stretch transformation to a logarithmic function, you need to multiply the function by a positive number. For example, if you have the function f(x) = log2 x, you can apply the vertical stretch transformation by multiplying it by 2, resulting in the function g(x) = 2log2 x.

Q: How do I apply the vertical shift transformation to a logarithmic function?

A: To apply the vertical shift transformation to a logarithmic function, you need to add a constant to the function. For example, if you have the function f(x) = log2 x, you can apply the vertical shift transformation by adding 2 to the function, resulting in the function g(x) = log2 x + 2.

Q: How do I apply the horizontal shift transformation to a logarithmic function?

A: To apply the horizontal shift transformation to a logarithmic function, you need to replace x with x-a, where a is a constant. For example, if you have the function f(x) = log2 x, you can apply the horizontal shift transformation by replacing x with x-2, resulting in the function g(x) = log2 (x-2).

Q: What are the key features of a logarithmic function after applying a transformation?

A: The key features of a logarithmic function after applying a transformation include:

Vertical Stretch: The function is stretched vertically by a factor of the transformation.
Reflection Across the x-axis: The function is reflected across the x-axis.
Vertical Shift: The function is shifted vertically by the transformation.
Horizontal Shift: The function is shifted horizontally by the transformation.

Q: How do I determine the type of transformation applied to a logarithmic function?

A: To determine the type of transformation applied to a logarithmic function, you need to analyze the function and identify the key features. You can use the following steps:

Identify the base of the logarithm: The base of the logarithm is the number that is raised to the power of the input.
Identify the exponent: The exponent is the number that is multiplied by the base.
Identify the transformation: The transformation is the operation that is applied to the function, such as reflection across the x-axis, vertical stretch, vertical shift, or horizontal shift.

By following these steps, you can determine the type of transformation applied to a logarithmic function and analyze its key features.