Introduction

Last time, we learned a whole lot about exponential functions. Now, we will explore how to rewrite exponential expressions in equivalent forms and manipulate them. We will cover different ways that will allow us to utilize the function in different ways.  

The Product Property

$b^{m+n}=b^m\cdot b^n$

The first major concept relates horizontal shifts to vertical stretches. It is the Product Property.

When we have an exponential function with a horizontal translation (a shift left or right), it takes the shape of $f\left(x\right)=b^{x+k}$. Here, $+k$ represents a shift left by $k$ units.

Similarly, if you have $f(x)=b^{x-k}$, $k$ represents a shift to the right by $k$ units. 

However, using the product property, we can rewrite this as $f\left(x\right)=b^k{b}^x$. Because $b^k$ is a constant, we can rewrite it as a constant, which we’ll call $a$. This turns the function into $f\left(x\right)=a{b}^x$. 

To reiterate, there is NO effect on the actual function’s behavior. They are the same function, we just redefined how our $k$ value was written.

Graphically, this rule implies that every horizontal translation of an exponential function is equivalent to some vertical dilation (stretch/compression).

The Power Property

The second rule relates horizontal stretches/compressions to changing the base. 

$(b^m{)}^n=b^{mn}$

What this rule is saying is that when we have an exponential function with a horizontal dilation ($f\left(x\right)=b^{cx}$), we can use the power property to group the base $b$ with the coefficient $c$ (note, we are using the reverse of this property: $b^{mn}=(b^m{)}^n$). This turns into $f\left(x\right)={\left(b^c\right)}^x$.

What this means is that every horizontal dilation of an exponential function is equivalent to some other form of an exponential function with a different base. 

The value of $c$ determines how many factors the dilation is. If $c=4$, the function is horizontally dilated by a factor of $4$.

Negative Exponents

The Negative Exponent Property, $b^{-n}=\frac{1}{b^n}=(\frac{1}{b}{)}^n$, is useful for simplifying exponential expressions. However, we can also use this property to change the base of an exponential expression.

Combining the Power Property and the Negative Exponents property, we can turn a function like $f\left(x\right)=b^{-n}$ into 

$f\left(x\right)=b^{-n}=b^{-1\cdot n}={\left(b^{-1}\right)}^n={\left(\frac{1}{b}\right)}^n$

Now, there are uses for this in simplifying expressions or combining terms. However, we also use this property to better identify Exponential Growth/Decay.

Unit Fractions

The rule is that $b^{\frac{1}{k}}=\sqrt[k]{b}$.

If $k$ is multiplied by a number greater than $0$ and less than $1$, it is a horizontal stretch, which is equivalent to taking the root to the base. Vice versa, if $k$ is multiplied by a number greater than $1$, it is a horizontal compression. However, this rule is primarily used for simplification purposes.

Practice Questions