Identify the equation that translates $$ y=\ln (x) $$ five units down. $$ y=\ln (x-5) $$ $$ y=\ln (x)+5 $$ $$ y=\ln (x+5) $$ $$ y=\ln (x)-5 $$

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To translate the function $$ y = \ln(x) $$ five units down, we need to subtract 5 from the entire function.

The general form for translating a function $$ f(x) $$ down by $$ k $$ units is given by:

$$
f(x) - k
$$

In this case, $$ k = 5 $$, so the translated function will be:

$$
y = \ln(x) - 5
$$

Now, let's identify the correct option from the given choices:

1. $$ y = \ln(x - 5) $$ - This translates the graph horizontally, not vertically.
2. $$ y = \ln(x) + 5 $$ - This translates the graph vertically upwards.
3. $$ y = \ln(x + 5) $$ - This also translates the graph horizontally.
4. $$ y = \ln(x) - 5 $$ - This is the correct translation downwards.

Thus, the correct equation that translates $$ y = \ln(x) $$ five units down is:

$$
y = \ln(x) - 5
$$
The correct equation is $$ y = \ln(x) - 5 $$.

Identify the equation that translates five units down.

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