The points (0,1)

, (1,14)
, and (2,196)
all lie on the line of the exponential function f(x)=14x
. On which logarithmic function do the points (14,1)
and (196,2)
lie?

1 answer

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:

  1. \((14, 1)\)
  2. \((196, 2)\)

If we assume that these points lie on a logarithmic function \(g(x)\), then we can express these relationships as:

  1. \(g(14) = 1\) implies that \(b^1 = 14\) or \(b = 14\).
  2. \(g(196) = 2\) 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:

\[ g(x) = \log_{14}(x) \]