Question

Choose the function that the graph represents.

Click on the correct answer.
y = f(x) = log(1/9)x
y = f(x) = loggx
y = f(x) = x9

Answer 1

`y=log_(9) (x)` (the middle choice)

Explanation:

Key Concepts

Concept 1. Exponential vs logarithm

Concept 2. Logarithm rules

Concept 1. Exponential vs logarithm

The first two choices are logarithmic functions whereas the last function is an exponential function. The graph cannot be that of an exponential function because exponential functions cannot cross the x-axis (an asymptote) unless a shift transformation is applied (which would look like adding or subtracting a constant number at the end of the equation.

A second way to verify is to simply input 2 into the function. The number 2 raised to the 9 power is 2*2*2*2*2*2*2*2*2=512, but the graph clearly does not have a height of 512 when the input is 2. Therefore, the correct answer cannot be the last choice.

Concept 2. Logarithm rules

One important rule for logarithms is that a number input into logarithm that matches the base of the logarithm will yield 1 as a result. In other words:

For all real numbers b, such that b is positive and not equal to 1,
`log_(b)(b)=1`

Observe that for the first option, this means that
`log_{(1)/(9)}((1)/(9))=1`. However, for an input of 1/9, the output is still below the x-axis -- a negative output -- clearly not 1.

Observe that for the second option, this means that
`log_(9)(9)=1`, and that for an input of 9, the output on the graph is at a height of 1.

Therefore, the correct function for this question must be the middle option.