Exponential Equation

• Introduction
• What is an Exponential Equation?
• Exponential Equations Property of Equality
• Conversion of Exponential Equations to Logarithmic Form
• Types of Exponential Equations
• Equations with Exponents
• Other Forms of Exponential Equations
• Real-life Applications of Exponential Equations
• Solved Examples
• Practice Problems

Introduction

An exponential equation is a mathematical equation in which the unknown variable appears in the exponent.

Exponential equations can also take various forms, such as exponential growth or decay equations, continuous exponential functions, and equations involving initial values.

Exponential equations are commonly used to model processes that exhibit exponential growth or decay, such as population growth, radioactive decay, compound interest, bacterial growth, and many other natural phenomena.

Solving exponential equations typically involves methods such as taking logarithms, using properties of exponents, or applying specific techniques depending on the form of the equation. These equations have widespread applications in science, engineering, finance, economics, and biology.

What is an Exponential Equation?

Exponential equations are mathematical expressions or equations where the variable appears as an exponent. In general, exponential equations can be represented as:

$$y = a \cdot b^x$$

Where:

$$y$$ is the dependent variable or output,

$$x$$ is the independent variable or input,

$$a$$ and $$b$$ are constants, with $$b$$ being the base of the exponential function.

Exponential equations can take various forms, including:

1. Exponential Growth: $$y = a \cdot (1 + r)^x$$

Where $$r$$ is the growth rate (expressed as a decimal).

2. Exponential Decay: $$y = a \cdot (1 - r)^x$$

Where $$r$$ is the decay rate (expressed as a decimal).

3. Continuous Exponential Growth/Decay: $$y = a \cdot e^{kx}$$

Where $$e$$ is Euler's number ($$e \approx 2.71828$$), $$a$$ and $$k$$ are constants, $$a$$ being the initial value and $$k$$ the growth constant.

4. Exponential Decay with Initial Value: $$y = a \cdot e^{-kx}$$

Where $$e$$ is Euler's number ($$e \approx 2.71828$$), $$a$$ and $$k$$ are constants, $$a$$ being the initial value and $$k$$ the decay constant.

Listed below are some examples of exponential equations:

1. $$3^x = 81$$: This equation is an exponential equation because the variable $$x$$ appears in the exponent. We can rewrite it as $$3^x = 81$$, where $$x$$ represents the exponent to which $$3$$ must be raised to obtain $$81$$.

2. $$5^{x - 3} = 625$$: This equation is an exponential equation, where $$x - 3$$ represents the exponent to which $$5$$ must be raised to obtain $$625$$.

3. $$64^{x - 7} = \sqrt{8}$$: This exponential equation involves radicals and we need to use the fact any radical can be written as a base with a fractional exponent to solve such equations.

Exponential Equations Property of Equality

If we have an equation of the form $$a^x = a^y$$, where $$a$$ is a non-zero real number and $$x$$ and $$y$$ are real numbers, then the exponents $$x$$ and $$y$$ must be equal. Mathematically, we can express this property as:

$$a^x = a^y \quad \text{if and only if} \quad x = y$$

This property essentially says that if two exponential expressions with the same base are equal to each other, then their exponents must be equal as well. This property is crucial for solving equations involving exponential functions, as it simplifies the process by allowing us to equate the exponents and solve for the variable.

For example, if we have the equation $$2^x = 2^3$$, according to the property you mentioned, we can conclude that $$x = 3$$, since the bases on both sides are equal.

Conversion of Exponential Equations to Logarithmic Form

Converting exponential equations to logarithmic form can be a useful technique for solving equations involving exponential functions. The process involves applying the properties of logarithms to rewrite the exponential equation in a form that makes it easier to solve. Here's how you can convert an exponential equation of the form $$a^x = b$$ to logarithmic form:

Given the exponential equation $$a^x = b$$, where $$a > 0$$, $$a \neq 1$$, and $$b > 0$$:

1. Take the logarithm of both sides: Use a logarithm with base $$a$$ or any other base to take the logarithm of both sides of the equation.

2. Apply the logarithmic property: According to the logarithmic property, $$\log_a(b^c) = c \cdot \log_a(b)$$. Apply this property to rewrite the equation.

3. Solve for the variable: Once the equation is in logarithmic form, you can solve for the variable $$x$$.

Here's the step-by-step process:

$$a^x = b$$

1. Take the logarithm of both sides:

$$\log_a(a^x) = \log_a(b)$$

2. Apply the logarithmic property:

$$x \cdot \log_a(a) = \log_a(b)$$

$$x \cdot 1 = \log_a(b)$$

$$x = \log_a(b)$$

So, the exponential equation $$a^x = b$$ can be rewritten in logarithmic form as $$x = \log_a(b)$$.

Types of Exponential Equations

We can categorize exponential equations based on the base of the exponential terms. Let's discuss each type:

1. Equations with the Same Bases on Both Sides:

In these equations, both sides have the same base raised to different powers. For example, $$4^x = 4^2$$ is an equation where the base $$4$$ is the same on both sides. To solve this type of equation, you can equate the exponents and solve for the variable $$x$$. We can equate the exponents using the fact that $$a^x = a^y$$ implies $$x = y$$.

2. Equations with Different Bases that Can be Made the Same:

These equations involve different bases, but one base can be expressed as a power of the other base. For example, $$4^x = 2^4$$ is an equation where the bases $$4$$ and $$2$$ are different, but $$4$$ can be expressed as $$2^2$$, making the bases the same. To solve this type of equation, you can rewrite the equation with the bases made the same and then solve for the variable $$x$$.

3. Equations with Different Bases that Cannot be Made the Same:

In these equations, the bases on both sides cannot be expressed as powers of each other. For example, $$4^x = 15$$ is an equation where the base $$4$$ and the constant $$15$$ cannot be related by the powers of each other. Solving these types of equations may involve using logarithms or other numerical methods.

For the first two types, we can manipulate the bases to simplify the equation, equate the exponents and solve for the variable. However, for the third type, more advanced techniques may be necessary to solve for the variable. Exponential and logarithmic are inverse operations and we extensively use this relation between exponential and logarithmic functions to solve the third type of equations.

To understand how to solve the problems of different scenarios of exponential equations lets see the examples of exponential equations:

1. Equations with the Same Bases on Both Sides:

Example 1. $$2^x = 2^3$$

Solution:

Since the bases are the same, we can equate the exponents: $$x = 3$$.

2. Equations with Different Bases that Can be Made the Same:

Example 2. $$4^x = 2^4$$

Solution:

Rewrite $$4$$ as $$2^2$$:

$$(2^2)^x = 2^4$$, which simplifies to $$2^{2x} = 2^4$$.

Now, equate the exponents:

$$2x = 4$$ and solve for $$x$$.

$$x = 2$$

3. Equations with Different Bases that Cannot be Made the Same:

Example 3. $$3^x = 5$$

Solution:

Since the bases are different and cannot be expressed as powers of each other, we typically use logarithms to solve.

Taking the logarithm of both sides, we get $$\log(3^x) = \log(5)$$.

By the logarithmic property, $$x\log(3) = \log(5)$$, so $$x = \frac{\log(5)}{\log(3)}$$.

On simplifying, we get $$x \approx 1.464$$.

Equations with Exponents

Equations with exponents are mathematical expressions where variables or constants are raised to a power. These equations can involve solving for the variable in the exponent or manipulating expressions with exponents. Below are more exponential equation examples that illustrate different scenarios and the methods used to solve them.

Example 1. $$2^x = 16$$

Solution:

Rewrite $$16$$ as $$2^4$$.

So, $$2^x = 16$$ becomes $$2^x = 2^4$$.

Equating the exponents, we get $$x = 4$$.

Example 2. $$3^{2x} = 81$$

Solution: Rewrite $$81$$ as $$3^4$$.

So, $$3^{2x} = 81$$ becomes $$3^{2x} = 3^4$$.

Equating the exponents, we have $$2x = 4$$.

Solving for x, we get $$x = 2$$.

Example 3. $$4^{x-1} = 8$$

Solution:

Rewrite $$4$$ as $$2^2$$ and $$8$$ as $$2^3$$.

So, $$4^{x-1} = 8$$ becomes $${(2^2)}^{x-1} = 2^3$$.

Applying the power to power rule of exponents, we get $$2^{2x-2} = 2^3$$.

Equating the exponents, we find $$2x - 2 = 3$$, which gives $$x = \frac{5}{2}$$.

Example 4. $$e^x = 20$$

Solution: To solve for $$x$$, we can take the natural logarithm of both sides:

$$\ln(e^x) = \ln(20)$$, resulting in $$x = \ln(20)$$.

Other Forms of Exponential Equations

Exponential equations can take various forms, and their solutions often depend on the specific context and structure of the equation. Here are some common forms of exponential equations along with their corresponding solution techniques:

1. Basic Exponential Equation:

$$y = a \cdot b^x$$

Solution Technique: To solve for $$x$$, take the logarithm of both sides, typically natural logarithm (ln) or base 10 logarithm (log), depending on the base of the exponential function.

2. Exponential Growth Equation:

$$y = a \cdot (1 + r)^x$$

Solution Technique: Similar to the basic exponential equation, take the logarithm of both sides to solve for $$x$$.

3. Exponential Decay Equation:

$$y = a \cdot (1 - r)^x$$

Solution Technique: Similar to the exponential growth equation, take the logarithm of both sides to solve for $$x$$.

4. Continuous Exponential Equation:

$$y = a \cdot e^{kx}$$

Solution Technique: Take the natural logarithm of both sides (ln) to solve for $$x$$.

5. Exponential Equation with Different Bases:

$$b_1^x = b_2^y$$

Solution Technique: If the bases are different, use logarithm properties to rewrite the equation and solve for one variable in terms of the other.

6. Exponential Equation with Variable Exponents:

$$b^{f(x)} = c$$

Solution Technique: If the exponent is a function of $$x$$, use appropriate algebraic techniques to isolate the exponent and solve for $$x$$.

7. Equations Involving Logarithmic and Exponential Functions:

Solution Technique: Equations combining exponential and logarithmic functions often require rearranging terms and using logarithm properties to solve for the variable.

Real-life Applications of Exponential Equations

Here are some some real-life applications of exponential equations:

1. Population Growth: The growth of populations, such as bacterial colonies, animal populations, or human populations, often follows exponential patterns. Exponential growth equations are used to model this growth, taking into account factors such as birth rates, death rates, and carrying capacity.

2. Compound Interest: Exponential equations are used in finance to model the growth of investments over time with compound interest. The equation $$A = P(1 + r)^t$$ represents the amount of money $$A$$ accumulated after $$t$$ years with an initial investment $$P$$ compounded annually at a fixed annual interest rate $$r$$.

3. Radioactive Decay: The decay of radioactive materials follows an exponential decay pattern. Exponential decay equations are used to model the rate at which radioactive isotopes decay over time, which is important in fields such as nuclear physics, radiology, and archaeology.

4. Epidemiology: In epidemiology, exponential equations are used to model the spread of infectious diseases through populations. The exponential growth of infections in the early stages of an outbreak can be described by equations that take into account factors such as the transmission rate and the population size.

5. Chemical Reactions: Exponential equations are used to model the rate of chemical reactions over time. Reaction rate equations often involve exponential terms that describe how the concentrations of reactants change as the reaction progresses.

6. Technological Growth: The adoption of new technologies often follows an exponential growth pattern. Exponential equations can be used to model the growth of technology adoption rates over time, which is important for forecasting trends and planning infrastructure development.

Solved Examples

Example 1. Solve for $$x$$: $$3^x = 27$$

Solution:

Since $$27 = 3^3$$, we can rewrite the equation as:

$$3^x = 3^3$$

Equating the exponents:

$$x = 3$$

So, the solution to the equation $$3^x = 27$$ is $$x = 3$$.

Example 2. Solve for $$x$$: $$5^{x+2} = 125$$

Solution:

Since $$125 = 5^3$$, we can rewrite the equation as:

$$5^{x+2} = 5^3$$

Equating the exponents:

$$x + 2 = 3$$

Solving for $$x$$:

$$x = 1$$

So, the solution to the equation $$5^{x+2} = 125$$ is $$x = 1$$.

Example 3. Solve for $$x$$: $$3^{x+4} = \frac{1}{9}$$

Solution:

Since $$9 = 3^2$$, we can write $$\frac{1}{9}$$ as $$\frac{1}{3^2}$$

Applying the rule for negative exponents, $$\frac{1}{3^2} = 3^{-2}$$.

So, $$3^{x+4} = \frac{1}{9}$$ becomes $$3^{x+4} = 3^{-2}$$

Equating the exponents:

$$x + 4 =-2$$

Solving for $$x$$:

$$x = -6$$

So, the solution to the equation $$3^{x+4} = \frac{1}{9}$$ is $$x = -6$$.

Example 4. Solve for $$x$$: $$\sqrt{5} = 25^{3x+4}$$

Solution:

We can write the radical $$\sqrt{5}$$ as $$5^{\frac{1}{2}}$$.

Since $$25 = 5^2$$, we can write $$25^{3x+4}$$ as $${(5^2)}^{3x+4}$$.

Applying the power to power rule of exponents, we can rewrite $${(5^2)}^{3x+4} = 5^{6x+8}$$.

So, $$\sqrt{5} = 25^{3x+4}$$ becomes $$5^{\frac{1}{2}} = 5^{6x+8}$$

Equating the exponents:

$$6x + 8 = \frac{1}{2}$$

Solving for $$x$$:

$$x = -\frac{5}{4}$$

So, the solution to the equation $$\sqrt{5} = 25^{3x+4}$$ is $$x = -\frac{5}{4}$$.

Example 5. Solve for $$x$$: $$2^x = 3$$

Solution:

Since the bases are different and cannot be expressed as powers of each other, we typically use logarithms to solve for $$x$$. Specifically, we'll use the natural logarithm (ln) to solve this equation.

Taking the natural logarithm of both sides:

$$\ln(2^x) = \ln(3)$$

Using the power property of logarithms:

$$x \cdot \ln(2) = \ln(3)$$

Dividing both sides by $$\ln(2)$$ to solve for $$x$$:

$$x = \frac{\ln(3)}{\ln(2)}$$

This gives us the value of $$x$$ as a decimal or in terms of natural logarithms.

So, the solution to the equation $$2^x = 3$$ is $$x \approx 1.5849625$$, or $$x = \frac{\ln(3)}{\ln(2)}$$.

Practice Problems:

Q1. Solve for $$x$$: $$10^{2x} = 100$$

1. $$x = 1$$
2. $$x = 2$$
3. $$x = 10$$
4. $$x = 100$$

Q2. Solve for $$x$$: $$2^x = 16$$

1. $$x = 16$$
2. $$x = 4$$
3. $$x = 2$$
4. $$x = 8$$

Q3. Solve for $$x$$: $$4^x = \sqrt{8}$$

1. $$x = \frac{4}{3}$$
2. $$x = \frac{2}{3}$$
3. $$x = \frac{3}{4}$$
4. $$x = \frac{3}{2}$$

Q4. Solve for $$x$$: $$3^x = 5$$

1. $$\frac{\ln(3)}{\ln(5)}$$
2. $$\frac{\ln(8)}{\ln(3)}$$
3. $$\frac{\ln(2)}{\ln(3)}$$
4. $$\frac{\ln(5)}{\ln(3)}$$

Q5. Solve for $$x$$: $$e^x = 8$$

1. $$\ln(8)$$
2. $$\ln(2)$$
3. $$\ln(4)$$
4. $$\ln(64)$$

Q1. What is an exponential equation?

Answer: An exponential equation is an equation in which the variable appears in the exponent. It typically takes the form $$a^x = b$$, where $$a$$ is the base, $$x$$ is the variable, and $$b$$ is a constant.

Q2. How do you solve exponential equations?

Answer: To solve exponential equations, you can use logarithms, exponential properties, or other algebraic techniques. The goal is to isolate the variable $$x$$ by manipulating the equation to a form where $$x$$ is no longer in the exponent.

Q3. What are the properties of logarithms used to solve exponential equations?

Answer: Some common properties of logarithms used in solving exponential equations include the product property, quotient property, and power property. These properties allow us to rewrite exponential expressions in a form that is easier to solve.

Q4. Can exponential equations have multiple solutions?

Answer: Yes, exponential equations can have multiple solutions, especially if the base is greater than 1 or if there are periodic functions involved. It's essential to check for extraneous solutions when solving exponential equations.

Q5. What are some real-world applications of exponential equations?

Answer: Exponential equations are commonly used to model growth and decay phenomena in various fields such as population growth, radioactive decay, compound interest, bacterial growth, and more. They are also prevalent in science, engineering, finance, and economics.

Q6. How do you know when to use logarithms to solve exponential equations?

Answer: Logarithms are typically used to solve exponential equations when the variable is in the exponent and cannot be isolated by other algebraic methods. If you encounter an equation where the variable is in the exponent and you cannot solve it directly, logarithms may be a suitable approach.

Q7. What is the difference between linear and exponential equations?

Answer: Linear equations involve variables raised to the power of 1, while exponential equations involve variables raised to various powers, including fractional or negative powers. Linear equations represent straight lines on a graph, whereas exponential equations represent curves that grow or decay exponentially.