Consider the function f(x)= log base 7 (x^2) such that x > 0. 1) Write a sequence of transformations that would transform f(x)= log base 7 (x^2) into the graph of f(x).

Question

2) Rewrite f(x) in terms of ln(x) and list the transformation(s) that are needed to transform y= ln (x) into f(x)

Answer 1

1. Sequence of Transformations

To determine the transformation, we first simplify the equation using the laws of logarithms.

Original: f(x)=log⁡7(x2)f(x)=log7​(x2)

Using the Power Property of Logarithms (log⁡b(mn)=n⋅log⁡b(m)logb​(mn)=n⋅logb​(m)):

f(x)=2⋅log⁡7(x)f(x)=2⋅log7​(x)

Comparing this to the parent function y=log⁡7(x)y=log7​(x), there is only one transformation:

1. Vertical Stretch by a factor of 2.

1. Rewrite in terms of ln⁡(x)ln(x) and Transformations

To rewrite the function using the natural logarithm (ln⁡ln), we use the Change of Base Formula: log⁡b(a)=ln⁡(a)ln⁡(b)logb​(a)=ln(b)ln(a)​.

Step 1: Apply Change of Base

f(x)=ln⁡(x2)ln⁡(7)f(x)=ln(7)ln(x2)​

Step 2: Apply Power Property

f(x)=2ln⁡(x)ln⁡(7)f(x)=ln(7)2ln(x)​

Step 3: Separate the constant to identify the transformation

f(x)=[2ln⁡(7)]⋅ln⁡(x)f(x)=[ln(7)2​]⋅ln(x)

Transformation from y=ln⁡(x)y=ln(x):

Because the term in the brackets is a constant multiplier greater than 1 (2ln⁡(7)≈1.028ln(7)2​≈1.028), the transformation is:

1. A vertical stretch by a factor of 2ln⁡(7)ln(7)2​.