Remember that exponential functions are named that because of the “\(x\)” in their exponents! “\(b\)” is called the base of the exponential function, since it’s the number that is multiplied by itself “\(x\)” times (and it’s not an exponential function when \(b=1\)).
\(b\) is also called the “growth” or “decay” factor.
It's an equation that has exponents that are $$ \red$$.
$$ \red 4^3 = \red 2^x $$ $$ \red 9^x = \red $$ $$ \left( \red \right)^ = \red 4^3 $$ $$ \red 4^ 1 = \red $$ In each of these equations, the base is different.
I always remember that the “reference point” (or “anchor point”) of an exponential function (before any shifting of the graph) is \((0,1)\) (since the “\(e\)” in “exp” looks round like a “”).
Exponential Problem Solving
(Soon we’ll learn that the “reference point” of a log function is \((1,0)\), since this looks like the “lo” in “log”).
When the base is greater than Remember again the generic equation for a transformation with vertical stretch \(a\), horizontal shift \(h\), and vertical shift \(k\) is \(f\left( x \right)=a\cdot k\) for exponential functions.
Remember these rules: When functions are transformed on the outside of the \(f(x)\) part, you move the function up and down and do “regular” math, as we’ll see in the examples below.
These types of equations are used in everyday life in the fields of Banking, Science, and Engineering, and Geology, as well as more fields.
We learned about the properties of exponents here in the Exponents and Radicals in Algebra Section, and did some solving with exponents here.