Exercise 46 Page 337

a The parent function is the simplest function within a family of functions. Examining the graph we see that it is a logarithmic function, and therefore the parent function is

y = lo g x .

b Examining the graph, we see that the asymptote is a vertical line through

x = 2 .

c When we add or subtract a constant to a function's input, the resulting transformation is a horizontal translation to the left and right respectively. In this case, we want a horizontal translation to the right by

2

units. This means we have to subtract

2

from

x .

To determine the base of the logarithm, we can use the one lattice point on the graph.

Let's substitute the known point on the graph into our function.

2 = lo g_{b} (6 - 2) \Leftrightarrow 2 = lo g_{b} 4

Using the definition of a logarithm, we can rewrite the equation.

2 = lo g_{b} 4 \Leftrightarrow b^{2} = 4

Let's solve for

b .

Notice that the base of a logarithm can only be positive.

b^{2} = 4

SqrtEqn

$LHS = RHS$

b = \pm 2

$b > 0$

b = 2

The base is

2 .

With this information, we can complete the equation.

y = lo g_{2} (x - 2)

Subchapter links

7.1.1 Using Logarithms to Solve Exponential Equations p.323

7.1.2 Investigating the Properties of Logarithms p.327-328

7.1.3 Writing Equations of Exponential Functions p.331-332

7.1.4 An Application of Logarithms p.336-337