Solving Exponential Equations

Exponential equations are equations where the variable appears as an exponent. Solving these equations involves various techniques depending on the structure of the equation. A common method is rewriting both sides of the equation with the same base and then equating the exponents. However, if the bases are different or not easily comparable, logarithms are often used to solve for the variable.

In this article, we will discuss multiple approaches to solve exponential equations, such as using the same base method and applying logarithmic properties.

What are Exponential Equations?

An exponential equation is an equation where the variable appears in the exponent. For example:

2^x = 8

In this equation, x is the exponent. Solving exponential equations involves isolating the variable in the exponent and often requires taking the logarithm of both the sides of the equation.

Definition of Exponential Functions

An exponential function is a mathematical expression in which a constant base is raised to the variable exponent. It can be written as:

f(x) = a^x

where a is a positive real number not equal to the 1 and x is any real number.

How to Use Logarithms to Solve Exponential Equations

When the bases of the exponents are different or the equation is more complex we can use logarithms to the solve exponential equations. Here’s a step-by-step guide:

Step 1: Rewrite the Equation:

Express the equation in the form a^x = b.

Step 2: Take the Logarithm:

Apply the logarithm to both THE sides of the equation. WE can use any logarithm base but commonly base 10 or base e are used.

log(a^x) = log(b)

Step 3: Apply Logarithm Properties:

Use the power rule of the logarithms log(a^x) = x ⋅ log(a).

x⋅ log(a) = log(b)

Step 4: Solve for the Variable:

Isolate x by dividing both the sides by log(a).

x = log(a)\log(b)

Step 5: Check the Solution:

The Substitute the value of the x back into the original equation to the verify the solution.

Properties of Logarithm

Some of the common properties of logarithm are listed in the following table:

Solved Examples on Exponential Equations

Example 1: Simple Exponential Equation

Problem: Solve 2^x = 32.

Solution:

Rewrite the Right-Hand Side: Express 32 as a power of 2.

32 = 2⁵

Set the Exponents Equal: Since the bases are the same set the exponents equal to the each other.

x = 5

Answer: x = 5

Example 2: Using Logarithms

Problem : Solve 3^x = 20.

Solution:

Take the Logarithm of Both Sides:

log(3^x) = log(20)

Apply the Power Rule of Logarithms:

x⋅log(3) = log(20)

Solve for x:

x = log(3)\log(20) ≈2.73

Answer : x≈2.73

Example 3: Natural Logarithms

Problem : Solve e^2x = 10.

Solution :

Take the Natural Logarithm of Both Sides:

ln(e^2x ) = ln(10)

Apply the Power Rule of Logarithms:

2x = ln(10)

Solve for x:

x = ln(10)\2≈1.15

Answer : x≈1.15

Example 4: Solving with Different Bases

Problem : Solve 5^2x = 125.

Solution:

Rewrite 125 as a Power of 5:

125 = 5³

Set the Exponents Equal:

2x = 3

Solve for x:

x = 2\3 = 1.5

Answer : x = 1.5

Example 5: More Complex Equation

Problem : Solve 2^{x + 1} = 8^{x − 2} .

Solution:

Rewrite 8 as a Power of 2:

8 = 2³

8^{x − 2} = (2³)^{x − 2} = 2^{3(x − 2)} = 2^{3x − 6}

Set the Exponents Equal:

x + 1 = 3x − 6

Solve for x:

x + 1 = 3x − 6

1 + 6 = 3x − x

7 = 2x

x = 2\7 = 3.5

Answer : x = 3.5

Practice Questions

Q 1. Solve 4^x = 64.

Q 2. Solve 10^2x = 1000.

Q 3. Solve e^x = 7.

Q 4. Solve 7^{x − 1} = 49.

Q 5. Solve 9^2x = 81.

Q 6. Solve 2^{x + 3} = 16.

Q 7. Solve 5^{x − 2} = 25.

Q 8. Solve 3^x = 81.

Q 9. Solve e^{x − 1} = 5.

Q 10. Solve 6^x = 36.

Answer Key

1. x = 3
2. x = 1.5
3. x ≈ 1.9459
4. x = 2
5. x = 1
6. x ≈ 3.7004
7. x ≈ 2.273
8. x = 4
9. x ≈ 2.6094
10. x = 2

Conclusion

Solving exponential equations requires an understanding of the exponential and logarithmic functions. By rewriting equations, applying logarithms and using the properties of exponents we can solve a variety of the exponential equations. The Practice with the different equations will help the solidify these concepts and improve problem-solving skills.

Also Check,

• Linear Equations
• Quadratic Equations
• Arithmetic Progression
• Coordinate Geometry
• Exponential Equations Worksheet