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.

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

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 sides of the equation.

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 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(b)/log(a)

Step 5: Check the Solution:

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

Solved Examples

Example 1: Solve the simple exponential equation 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: Use logarithms to 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; 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; 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 = 7 /2= 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