Final answer:
To create the function rule for g(x), which translates f(x) = log(x) up by 10 units and right by 1 unit, the rule would be g(x) = log(x - 1) + 10. The translated graph of g(x) will be the same as the graph of f(x), but shifted accordingly to the translation parameters.
Step-by-step explanation:
To write the function rule for g(x), which is a translation of the function f(x) = log(x) 10 units up and 1 unit right, we need to adjust the original function to reflect these changes. A vertical shift of 10 units up is represented by adding 10 to the function, and a horizontal shift of 1 unit right is represented by subtracting 1 from the x variable inside the log function. Therefore, the new function, g(x), would be represented as g(x) = log(x - 1) + 10.
To verify our translation, we can consider the characteristic of the logarithm function. For instance, the natural log of 10 is approximately 2.30, indicating the rate at which the logarithm function grows. If we were to graph this function, we would label the horizontal axis as x and the vertical axis as f(x) or g(x), depending on the function being represented. The function f(x) = log(x) would have a curve increasing from the left to right, and the translated function g(x) would show the same curve but shifted to the right by 1 and up by 10.
When using a calculator to find logarithms, you simply press the log button to obtain the common logarithm, and to reverse this process, you take the inverse log or calculate 10 to the power of the logarithm. This functionality is essential for verifying the points on the graph or calculating specific values for f(x) or g(x).