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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.

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.

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.