The Meaning of Solving Exponential Equations

Intersection point of functions

In chapter 3, we learned that the meaning of solving an equation is to find the intersection point(s) between two functions. The intersection point(s) between the graphs of any two functions $f(x)$ and $g(x)$ can be found algebraically by setting the two functions equal to each other:

$f(x)=g(x)$

When the functions are equal, the value of $x$ is the same for both functions, as is the function value. In other words, $f(x)=g(x)$ means that the two functions have the same input $x$ as well as the same output (i.e $f(x)=g(x)$). For example, solving the equation $32\left(2^{x+3}\right)=64$ means finding the intersection point between the two functions $f(x)=32\left(2^{x+3}\right)$ and $g(x)=64$ (figure 1).

FIgure 1. Intersection of $y=f(x)$ and $y=g(x)$

Finding the $x$-value given a function value

Another interpretation of the equation $(32)2^{x+3}=64$ is finding the $x$ value when the function value of $f(x)=(32)2^{x+3}$ is 64 (figure 2).

Figure 2. Meaning of solving an equation

Graphically, this means finding the function value as a given $y$-value on a graph, then moving vertically down to the $x$-axis to determine the corresponding $x$-value. In figure 2, to solve $(32)2^{x+3}=64$, we graph the function $f(x)=(32)2^{x+3}$, look for 64 on the $y$-axis then determine which $x$-value has a function value at 64. In this case, $x=-2$.

Solving an equation in one variable

Algebraically, $(32)2^{x+3}=64$ is an equation in one variable. When solving an equation in one variable, we find the value of the variable that satisfies the equation (e.g., the $x$-value). There is no function value to report as the equation is in just one variable.

For example, when solving the equation, $3^x=9$, we find the value of $x$ that makes the equation true. The value of $x$ is 2 because $3^2=9$. We need algebraic methods to solve equations, including the properties of equality covered in chapter 3.

In summary, when we set functions equal to each other, we are finding the intersection point between two functions (e.g., $f(x)=3^x$ and $g(x)=9$). In this example, the intersection point between the graphs of the two functions $f(x)$ and $g(x)$ is (2, 9) because the $y$-value is the function value $f(2) = 3^2 = 9$ and $g(x)=9$. However, when we are solving an equation algebraically, there is no function in sight so we can just report the value of the variable (e.g., $x$).

Solving Exponential Equations Using the Property of Equality

We may use the property of equality for solving an exponential equation if the exponential expressions on each side of the equation have the same base.

For example, to solve the equation $3^x = 9$, we can write $9 = 3^2$ so that $3^x = 3^2$. The property of equality then tells us that $x = 2$ because with the same base, the exponents must be equal. Although it may at first appear that the exponential expressions in an equation do not have the same base, they can often be transformed to expressions with the same base.