To determine the logarithmic function on which the points (14,1) and (196,2) lie, we should first recognize that logarithmic functions can be expressed in the form:
\[ g(x) = \log_b(x) \]
where \(b\) is the base of the logarithm.
We have two points:
- \((14, 1)\)
- \((196, 2)\)
If we assume that these points lie on a logarithmic function \(g(x)\), then we can express these relationships as:
- \(g(14) = 1\) implies that \(b^1 = 14\) or \(b = 14\).
- \(g(196) = 2\) implies that \(b^2 = 196\).
Next, let's find \(b\) from the second point:
\[ b^2 = 196 \]
Taking the square root of both sides:
\[ b = \sqrt{196} = 14 \]
Thus, both points give us the same base \(b = 14\).
Therefore, the logarithmic function that contains both points (14, 1) and (196, 2) is given by:
\[ g(x) = \log_{14}(x) \]