# Natural Logarithms

## Solved Problems

### Example 3.

Sketch the graph of the function \[y = \ln \left( {x + 1} \right) - 1.\]

Solution.

The graph of the function \(y = \ln \left( {x + 1} \right) - 1\) is that of \(y = \ln x\) shifted one unit to the left (we obtain the graph of the function \(y = \ln \left( {x + 1} \right)\)) and one unit down \(\left({\text{Figure } 2}\right).\)

### Example 4.

Sketch the graph of the function \[y = \left| {\ln x} \right|.\]

Solution.

The graph of the function (Figure \(3\)) can be built in the result of the following transformations. The portion of the graph of \(y = \left| {\ln x} \right|\) lying at \(x \ge 1\) is identical to the graph of \(y = \ln x,\) while the portion \(y \lt 0\) at \(0 \lt x \lt 1\) is reflected about the \(x\)-axis into the upper half-plane.

### Example 5.

Sketch the graph of the function \[y = \left| {\ln \left| x \right|} \right|.\]

Solution.

First we get the graph of \(y = \left| {\ln x} \right|\) as described in the previous example. Then we reflect the graph of the function about the \(y\)-axis to the left half-plane to obtain the graph of the function \(y = \left| {\ln \left| x \right|} \right|\) (Figure \(4\)).