As the name indicates, exponential equations are functions that have exponents. An exponent indicates how many times to multiply the base by itself. When the exponent has a variable and the function is in an equation, it is called an exponential equation. In this article, we are going to discuss exponential functions and the various formulas that relate to them. We will also discuss growth and decay equations, as well as real-world applications of exponential equations. In order to solidify your understanding, we will go through some examples to illustrate these concepts.

What are Exponential Equations?

An exponential equation is a mathematical equation that involves an exponent. For instance: \(5^{x-2}=625\) is an exponential equation with power of 5 in variable x – 2.

It represents a relationship where a variable appears as the exponent of a base. The general form of an exponential equation is \(y = a \times b^x\), where \(a\), \(b\), and \(x\) are variables or constants.

Exponential equations are widely used to model phenomena that involve exponential growth or decay, such as population growth, compound interest, radioactive decay, and more.

They help us understand and predict how quantities change over time based on certain growth or decay factors.

Exponential Equation Real Example

Let's consider the example of a bacterial colony.

Suppose we have a bacterial colony that doubles in size every hour. We start with an initial population of 100 bacteria.

To model the growth of the colony using an exponential equation, we can use the formula:

\(y = a \times 2^x\)

Here, \(a\) represents the initial population (100 bacteria), \(2\) is the growth factor (as the colony doubles in size), and \(x\) denotes the number of hours that have passed.

Let's say we want to find the population after 5 hours. We can plug in the values into the equation:

\(y = 100 \times 2^5\)

Evaluating the equation, we find:

\(y = 100 \times 32\)

Thus, after 5 hours, the bacterial colony would have a population of 3,200 bacteria.

This exponential equation helps us understand the rapid growth of the bacterial colony over time, with the population doubling every hour.

Types of Exponential Equations

Exponential equations come in different forms and are widely used in mathematics and scientific disciplines. Let's explore three main types:

Exponential Growth Equations: These equations describe situations where a quantity experiences rapid growth over time. They can be represented as \(y = a \times b^x\), where \(a\) represents the initial value, \(b\) is the growth factor, and \(x\) is the independent variable.

Exponential Decay Equations: These equations illustrate scenarios where a quantity gradually decreases over time. They follow the format \(y = a \times (1 - b)^x\), with \(a\) denoting the initial value, \(b\) representing the decay factor (typically between 0 and 1), and \(x\) serving as the independent variable.

Exponential Growth and Decay Equations with a Constant Base: In some cases, exponential growth and decay equations utilize a constant base \(e\), approximately 2.718. They can be expressed as \(y = a \times e^{kx}\), where \(a\) signifies the initial value, \(k\) represents the growth or decay rate, and \(x\) remains the independent variable.

Understanding these three types of exponential equations is essential for analysing the dynamics of exponential growth and decay in various mathematical and scientific contexts.

Exponential Equation Formula

We can define the exponential function using a formula, \(f\left(x\right)=a^x\). The curve of the exponential function depends particularly on the value of x.

Mathematically, any function of the form \(f\left(x\right)=a^x\) is termed as an exponential function.

Here, a>0 and cannot be equal to 1.

Also, x is a real number.

However, when the variable is negative, the function remains undefined for -1<x<1.

In such cases, ‘x’ is the variable and ‘a’ is the base of the exponential function.

Exponential Equations to Logarithmic Form

To convert an exponential equation \(y = a \times b^x\) into logarithmic form, we use the logarithm base \(b\) on both sides.

The resulting logarithmic equation is \(\log_b(y) = x\). Here, \(\log_b(y)\) represents the logarithm of \(y\) to the base \(b\).

This allows us to convert equations with exponential expressions into equations involving logarithms. By doing so, we can simplify calculations, solve for unknowns.

How to Solve Exponential Equations

While solving exponential equations we might come across two different scenarios. These can be exponential functions with the same bases and different bases.

Let us discuss each of the above mentioned cases with example:

Solving Exponential Equations with Same Bases

In some scenarios, the exponential equations may have the same base on both sides of an equation. For instance: \(5^x=5^3\) has the same base 5.

In these cases, when we have to solve the equations, we simply need to apply the property of equality of exponential functions. Using this property, we set the exponentials to be the same and solve the variable.

Let us take an example to solve such equations:

Example: Solve for y \(7^{y+2}=343^y\)

Solution:

We know that \(7^3=343\), so the given equation can be written as:

\(7^{y+2}=\left(7^3\right)^y\)

\(7^{y+2}=7^{3y}\)

As the bases are same, so we can equate the powers:

\(y+2=3y\)

\(2y=2\)

y = 1.

Solving Exponential Equations with Different Bases

There might be some exponential equations where the bases are not the same and cannot be made the same using different operations. In these cases, the exponential equations are solved using logarithms. For instance: \(5^x = 7\), we can see that the exponent functions on both sides of the equation do not have the same base and the base cannot be made the same even after using operations.

So, we can solve such equations as:

• We need to convert the exponential function to logarithmic form using the formula \(b^x=a\ \Rightarrow \log _ba=x\) and then solve the variable.
• We need to apply log on both sides of the equation and then solve for the variable. Here we use the property \(a^m=m\log a\).

Let us solve \(5^x=3\) using each of the above mentioned method:

Method 1:

Let us convert \(5^x=3\) into logarithmic form.

We get:

\(\log _53=x\)

Using the change of base property:

\(x=\frac{\log 3}{\log 5}\)

Method 2:

In this method let us take log on both the sides of the given equation:

\(\log 5^x=\log 3\)

Using the property of \(a^m=m\log a\) on LHS of the above equation:

\(x\log 5=\log 3\)

Or, \(x=\frac{\log 3}{\log 5}\)

Exponential Function Graph

Let us understand the exponential function from the graph given below. It can be seen that as the exponent increases, the curve gets steeper and the rate of growth increases.

So, for x> 1, the value for \(y\ =\ f_n\left(x\right)\) increases as the value of n increases.

From the above graph we can see that the nature of the curve is largely dependent on the degree of the exponent. For a polynomial with higher degree, the growth will be automatically high. The domain for any exponential function will be all real numbers, whereas the range will be all the positive reals.

Exponential Function Property

For a function, the exponential graph of the function represents its properties.

Let us understand this using the exponential function \(y=2^x\)

Graphically, we can represent \(y=2^x\) as:

When the base if greater than 1, the properties for such exponential functions are:

• The graph passes through the point (0,1) on the cartesian plane.
• The domain of such a graph is all real numbers.
• The range for such a function is always y>0.
• This function forms an increasing graph.
• Also, such a graph is asymptotic to the negative x-axis, as x approaches negative infinity.
• The graph increases boundlessly as x approaches to positive infinity.
• The graph is smooth and continuous.

Let us take another exponential function \(y=2^{-x}\).

Graphically, the function \(y=2^{-x}\) can be represented as:

When the base of the exponential function lies between 0 and 1, it has the following properties:

• The graph passes through the point (0,1) on the cartesian plane.
• The domain of such a graph is all real numbers.
• The range for such a function is always y>0.
• This function forms a decreasing graph.
• Also, such a graph is asymptotic to the x-axis, as x approaches to positive infinity.
• The graph increases boundlessly as x approaches to negative infinity.
• The graph is smooth and continuous.

Exponential Growth

For a function experiencing exponential growth, the quantity experiences a slow increase at first that increases rapidly thereafter.

The rate of change increases over time and becomes faster as the time passes. The formula that defines the exponential growth is:

\(y=\left(1+r\right)^x\)

Here, ‘r’ is the growth percentage.

Exponential Decay

For a function experiencing exponential decay, the quantity experiences a slow decrease at first that decreases rapidly thereafter. The rate of change decreases over time and becomes slower as the time passes. The formula that defines the exponential decay is:

\(y=\left(1-r\right)^x\)

Here, ‘r’ is the decay percentage.

Exponential Equation Theorem

When we need to solve exponential equation, the following theorem is very useful:

Statement: If ‘a’ is a non-zero constant and \(a^x=a^y\) then, x = y.

Proof: We are given that:

\(a^x=a^y\), here, a is not equal to 1.

\(\frac{a^x}{a^y}=1\)

\(a^{x-y}=1\)

x – y = 0

x = y

Hence Proved

Exponential Function Derivative

Let us understand the derivative of exponential function:

In simple terms, the derivative of a function \(e^x\) with respect to x is \(e^x\). We can also write it as: \(\frac{d}{dx}e^x=e^x\)

Or, we can say that the exponential function has special property, which says that the derivative of an exponential function is the function itself.

Exponential Equation Solved Examples

Q1. The integral form of exponential growth equation will be

Solution:

The integral form of the exponential growth equation is obtained by integrating the differential equation that represents exponential growth.

For an exponential growth equation of the form [latex]\frac{{dy}}{{dt}} = k \cdot y[/latex], where [latex]y[/latex] represents the quantity at time [latex]t[/latex] and [latex]k[/latex] is a constant growth rate, the integral form is:

[latex]\int \frac{{dy}}{{y}} = \int k , dt[/latex]

Integrating both sides, we get:

[latex]\ln|y| = kt + C[/latex]

Here, [latex]\ln[/latex] represents the natural logarithm, [latex]C[/latex] is the constant of integration.

By exponentiating both sides, we can express the integral form in terms of the original variable [latex]y[/latex]:

[latex]y = e^{kt+C}[/latex]

Simplifying further, we have:

[latex]y = C \cdot e^{kt}[/latex]

The constant [latex]C[/latex] represents the initial value of the quantity [latex]y[/latex].

This integral form of the exponential growth equation provides a general expression for the quantity [latex]y[/latex] as a function of time [latex]t[/latex], considering a constant growth rate [latex]k[/latex].

Q2. Solve \(\frac{27}{3^{-x}}=3^6\)

Solution: 

We know that \( 27 = 3^3\)

So, the equation above can be written as:

\(\frac{3^3}{3^{-x}}=3^6\)

Also we know that:

\(\frac{a^m}{a^n}=a^{m-n}\) using quotient property of exponents.

So,

As the bases are same, so the powers will be equal:

3 + x = 6

x = 3

Q3. Simplify \(2^x-2^{x+1}\)

Solution:

Given exponential function: \(2^x – 2^{x+1}\)

By using the property: \(a^x \times a^y = a^{x+y}\)

Hence, \(2^{x+1}\) can be written as \(2^x. 2\)

Thus the given function is written as:

\(2^x-2^{x+1} = 2^x-2^x. 2\)

Now, factor out the term \(2^x\)

\(2^x-2^{x+1} = 2^x-2^x. 2 = 2^x(1-2)\)

\(2^x-2^{x+1} = 2^x(-1)\)

\(2^x-2^{x+1} = – 2^x\)

Therefore, the simplification of the given exponential function \( 2^x – 2^{x+1}\) is \( – 2^x\).

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