Answer to Question #151463 in Algebra for Patrick Joncourt

Question

Reflect on the concept of exponential and logarithm functions. What concepts (only the names) did you need to accommodate these new concepts in your mind? What are the simplest exponential and logarithmic functions with base b ≠ 1 you can imagine? In your day to day, is there any occurring fact that can be interpreted as exponential or logarithmic functions? What strategy are you using to get the graph of exponential or logarithmic functions?

Answer 1

Reflecting on the concept of exponential and logarithm functions, takes me to the basics. These functions are the opposite of each other. Logarithm function (y = log_ax) is the inverse equivalent of the (x = a^y) exponential function. With a parent exponent f(x) = a^x always has a horizontal line at y=0, unless when a =1. Positive numbers cannot be raised to a negative number.  

The simplest exponential and logarithmic functions with base b ≠ 1 are, y=b^x and y= log _bx respectively. Exponential function f(x) = (8)^x has a base of b=8. In our day to day, there a many occurring real-world situations that can be interpreted as exponential or logarithmic functions. These can be used as function models to compute investment, bacterial culture, carbon dating, population growth and many other complex numeric calculations.

One of my strategies I use to get the graph of exponential or logarithmic functions, is to start with a table. I think of my table columns as an input and output. I do this by creating a t-chart with x and y to represent my columns. I then, follow with my inputs/outputs from my equation. This is a very powerful method that can help in graphing any type of exponential or logarithmic functions.

 

Graphing:

This is how I perceive tables, and I like to think of it as an input/output system. Starting with an equation like y=2^x, when you input 0 for x, then y would be 1. This essentially means that this graph passes through point (0, 1). When creating a table, you want to create a t-chart and put x and y on top. Put 0 under x, and 1 under y. If you plug in 1 for x, then you get 2 for y. Again, you would put 1 under x, and 2 under y. This pattern can go on forever, and works for any type of graph. Hope this helps!