Logarithms

Question

1. Use the properties of logarithms and transformations to describe how the graph of g(x)=ln(e^²x)

2. Simplify: log(1-1/x)-log(1+1/x)

 

I need help with these two problems. Could you help explain how to solve them? Also, please note that "1/x" in the second problem is a fraction. 

Thanks so much!!!

Answer 1

Let's review the properties of logarithms you'll need:

 

Problem 1:  ln(ab) = ln(a) + ln(b)

                  ln(a^b) = b*ln(a)

                  log_a(a) = 1 (for this problem, ln(e) = 1)

 

Problem 2:  ln(a/b) = ln(a) - ln(b)

 

To solve problem 1, Use the first property with a = e² and b = x.  Now, use the 2nd and 3rd properties in sequence.  You will end up with a function of the form g(x) = ln(x) + C, which indicates a vertical translation of the fundamental function y = ln(x).

 

To solve problem 2, apply the property in reverse.  You will end up with a complex fraction inside of a logarithm.  Simplify the fraction.