# Domain And Range Of Logarithmic Functions From Equation Worksheet

## 6 problems

Understanding the domain and range of logarithmic functions from an equation involves identifying permissible inputs (domain) and resulting outputs (range). For $$y = \log_b(x)$$, the domain is $$x > 0$$ since logarithms are only defined for positive numbers, and the range is all real numbers ($$y \in \mathbb{R}$$). Reviewing a domain and range of logarithmic functions from equation examples and practicing with domain and range of logarithmic functions from equation questions solidifies these concepts.

Algebra 2
Exponential Functions

## How Will This Worksheet on "Domain and Range of Logarithmic Functions from Equation" Benefit Your Student's Learning?

• Helps students grasp what logarithms are and how they work.
• Enhances skills in analyzing equations to determine permissible inputs (domain) and possible outputs (range).
• Makes it easier to solve real-world problems, like how things grow or shrink over time.
• Promotes critical thinking as students analyze how changes in the function's parameters affect its domain and range.
• Strengthens understanding of algebraic transformations affecting domain and range such as shifts and stretches.

## How to Domain and Range of Logarithmic Functions from an Equation?

• Recognize the logarithmic function in the form $$y = \log_b(x)$$, where $$b > 0$$ and $$b \neq 1$$.
• Logarithmic functions are defined for $$x > 0$$ because the logarithm of zero or a negative number is undefined in real numbers.
• The range of $$y = \log_b(x)$$ includes all real numbers ($$y \in \mathbb{R}$$) because logarithmic functions can yield any real value depending on the input $$x$$.
• Recognize how changes in the base $$b$$ and any transformations affect the domain and range, such as shifts left or right on the x-axis and vertical stretches or compressions on the y-axis.

## Solved Example

Q. What is the domain of this logarithmic function?$\newline y=\log_9(x+1)+3\newline$Express the domain in inequality notation.
Solution:
1. Identify Argument: Identify the argument of the logarithm.$\newline$Argument: $x + 1$
2. Set Inequality: Set the argument greater than $0$.$\newline$$x + 1 > 0$
3. Solve Inequality: Solve the inequality.$\newline$$x > -1$
4. Write Domain: Write the domain in interval notation.$\newline$Domain: $x > -1$ or $(-1, \infty)$

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