Final answer:
The function g(x) for a 6-unit downward and 3-unit leftward translation of f(x) = log(x) is g(x) = log(x + 3) - 6. This modifies the graph of the original function to reflect the desired shifts.
Step-by-step explanation:
Translation of the Logarithmic Function
To perform a translation of the function f(x) = log(x), we need to adjust its rule to reflect the given transformations. A translation of 6 units down means we subtract 6 from the function's output. A translation of 3 units to the left means we need to input a value that is 3 units higher to get the original output. Therefore, we need to replace x with (x + 3) in the function. The combination of these two transformations gives us the function rule for g(x):
g(x) = log(x + 3) - 6
This function will graph the original log function, but every point will be moved 3 units to the left along the x-axis and 6 units down along the y-axis.
Some important properties of logarithms that support this process are:
- The logarithm of a product is the sum of the logarithms.
- The logarithm of a quotient is the difference between the logarithms.
- The logarithm of an exponentiated number and its base results in the exponent.
- Transformations in the argument of the logarithm (inside the log) affect the x-values in the graph, while transformations outside the log function affect the y-values.