Identify the equation that translates $y=\ln (x)$ five units down.
We are asked to identify the equation that represents a vertical translation of the function \(y = \ln(x)\) five units downward. A vertical shift of a function \(f(x)\) by a constant \(k\) results in a new function \(g(x) = f(x) + k\). If \(k > 0\), the graph shifts upward; if \(k < 0\), the graph shifts downward. Here, we are shifting the graph downward by 5 units, so \(k = -5\).
Applying the rule for vertical translation to the given function \(f(x) = \ln(x)\), we substitute \(k = -5\) into the general form \(g(x) = f(x) + k\). This gives the new function:
\[y = \ln(x) + (-5) = \ln(x) - 5\]
This confirms that subtracting a constant from the entire function shifts its graph down by that constant amount.
We compare our result, \(y = \ln(x) - 5\), with the given choices:
The correct option is clearly the fourth one.
In engineering, such vertical shifts are common when calibrating sensors. For instance, if a sensor measuring a logarithmic growth process has a baseline drift, you might need to adjust its output equation by subtracting a constant value to re-center the readings at zero. Similarly, in economics, shifting a demand curve vertically by a fixed amount representing a tax or subsidy per unit is a fundamental application of function translations.