The points (0,1),(1,14), and (2,196) all lie on the line of the exponential function f(x)=14

Question

x
.
On which logarithmic function do the points (14,1) and (196,2) lie?
A. g(x)=x
14

B. g(x)=log
14
​
x
C. g(x)=log14
x

D. g(x)=logx

Answer 1

To determine the logarithmic function on which the points \((14,1)\) and \((196,2)\) lie, we can use the relationship between exponential functions and logarithmic functions.

Given the exponential function \(f(x) = 14^x\):

1. If \(f(1) = 14\), then \(g(14) = 1\) because logarithms give you the exponent needed to reach a certain value: \[ g(14) = \log_{14}(14) = 1 \]

2. If \(f(2) = 196\), then \(g(196) = 2\) because: \[ g(196) = \log_{14}(196) \]

To express \(196\) in terms of the base \(14\):

We know: \[ 196 = 14^2 \]

Thus: \[ g(196) = \log_{14}(196) = \log_{14}(14^2) = 2 \]

Now, we can identify that the logarithmic function that the points \((14,1)\) and \((196,2)\) lie on is:

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

Therefore, the correct answer is \(B\) or \(g(x) = \log_{14}(x)\).