Problem 82 - Algebra 2

Question

Find the domain and the range of each function. $$ y=\log _{8} x-2 $$

Step-by-Step Solution

Answer

The domain of the logarithmic function \(y=\log_8 x-2\) is \(x>0\) and its range is \(-\infty < y < \infty\).

Step 1: Find the Domain

The function is defined for only the positive values of x in the natural log function. Therefore, when the base is 8, the rule remains the same. Since x must be greater than zero, the domain of the function \(y=\log_8 x-2\) is \(x>0\).

Step 2: Find the Range

The range of a basic logarithm function is all real numbers(\(-\infty,

Key Concepts

Domain of a Function

The domain of a function tells us which input values are valid for the function. For logarithmic functions, the input value is the number inside the log. Generally, logarithms are only defined for positive numbers. This means, for any logarithmic function expressed as \(y = \log_b(x)\), the domain must be such that \(x > 0\).

In our exercise, the function is \(y = \log_8(x) - 2\). The logarithm \(\log_8(x)\) is only defined when \(x > 0\). Thus,

• The domain for this function is all positive values of \(x\).
• We write this as \(x > 0\).

This characteristic remains consistent across all logarithmic functions regardless of base since you cannot take the logarithm of a non-positive number. Understanding this ensures you can correctly identify the domain for any logarithmic function.

Range of a Function

The range of a function refers to all possible values that output can take. For basic logarithmic functions like \(y = \log_b(x)\), the range is all real numbers. This means the function can output any real number value, spanning from \(-\infty\) to \(\infty\).

In the function \(y = \log_8(x) - 2\), there is a downward shift of the entire function by 2 units. This transformation affects only the position of the graph but does not change the range. Therefore, the range of this function is:

• Still all real numbers, or \(-\infty < y < \infty\).

Logarithmic functions inherently have this range because their graph extends infinitely in both the positive and negative directions vertically, capturing any real number within its output.

Transformation of Functions

Transformations alter the position or shape of a function's graph. They can occur due to shifts, stretches, or reflections on the axes. In our function \(y = \log_8(x) - 2\), there is a transformation evident as a vertical shift.

Vertical shifts involve adding or subtracting values outside the logarithm:

• For this function, subtracting 2 means the graph of the basic function \(\log_8(x)\) shifts 2 units downward.

This transformation does not affect the domain (which remains \(x > 0\)), but it influences where the graph is situated on the coordinate plane. While the domain and range stay consistent, vertical shifts move the graph up or down without changing its essential properties.

Understanding transformations enables you to predict how a graph will behave when subjected to alterations, helping you in visualizing and solving logarithmic functions effectively.

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Problem 82

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Problem 83

Other exercises in this chapter

Problem 82

Solve each equation. If necessary, round to the nearest ten-thousandth. $$ \frac{1}{2} \log x+\log 4=2 $$

View solution

Problem 82

Expand each logarithm. \(\log _{b} \frac{\sqrt{x} \sqrt[3]{y^{2}}}{\sqrt[5]{z^{2}}}\)

View solution

Problem 83

Solve each equation. If necessary, round to the nearest ten-thousandth. $$ 4 \log _{3} 2-2 \log _{3} x=1 $$

View solution

Problem 83

Expand each logarithm. \(\log _{4} \frac{\sqrt{x^{5} y^{7}}}{z w^{4}}\)

View solution

Key Concepts

Domain of a Function

The domain of a function tells us which input values are valid for the function. For logarithmic functions, the input value is the number inside the log. Generally, logarithms are only defined for positive numbers. This means, for any logarithmic function expressed as \(y = \log_b(x)\), the domain must be such that \(x > 0\).

In our exercise, the function is \(y = \log_8(x) - 2\). The logarithm \(\log_8(x)\) is only defined when \(x > 0\). Thus,

• The domain for this function is all positive values of \(x\).
• We write this as \(x > 0\).

This characteristic remains consistent across all logarithmic functions regardless of base since you cannot take the logarithm of a non-positive number. Understanding this ensures you can correctly identify the domain for any logarithmic function.

Range of a Function

The range of a function refers to all possible values that output can take. For basic logarithmic functions like \(y = \log_b(x)\), the range is all real numbers. This means the function can output any real number value, spanning from \(-\infty\) to \(\infty\).

In the function \(y = \log_8(x) - 2\), there is a downward shift of the entire function by 2 units. This transformation affects only the position of the graph but does not change the range. Therefore, the range of this function is:

• Still all real numbers, or \(-\infty < y < \infty\).

Logarithmic functions inherently have this range because their graph extends infinitely in both the positive and negative directions vertically, capturing any real number within its output.

Transformation of Functions

Transformations alter the position or shape of a function's graph. They can occur due to shifts, stretches, or reflections on the axes. In our function \(y = \log_8(x) - 2\), there is a transformation evident as a vertical shift.

Vertical shifts involve adding or subtracting values outside the logarithm:

• For this function, subtracting 2 means the graph of the basic function \(\log_8(x)\) shifts 2 units downward.

This transformation does not affect the domain (which remains \(x > 0\)), but it influences where the graph is situated on the coordinate plane. While the domain and range stay consistent, vertical shifts move the graph up or down without changing its essential properties.

Understanding transformations enables you to predict how a graph will behave when subjected to alterations, helping you in visualizing and solving logarithmic functions effectively.