Solving Exponential Equations With Logarithms Answer Key

Solving Exponential Equations with Logarithms: A Comprehensive Guide

Exponential equations, those containing variables in the exponent, often require logarithmic functions for their solution. This guide provides a comprehensive walkthrough of solving exponential equations using logarithms, covering various scenarios and offering detailed explanations with example problems and answers. We'll explore different logarithmic properties and techniques to effectively tackle diverse exponential equations.

Understanding Exponential Equations

Before diving into the solutions, let's solidify our understanding of exponential equations. They are equations where the variable appears in the exponent. A common form is:

a^x = b

where 'a' is the base, 'x' is the exponent (our variable), and 'b' is the result. Solving for 'x' requires the use of logarithms.

Introducing Logarithms: The Key to Unlocking Exponents

Logarithms are the inverse functions of exponential functions. They allow us to bring down the exponent and solve for the variable. The basic logarithmic equation is:

log_ab = x

This reads as "the logarithm of b to the base a is x". It's equivalent to the exponential equation a^x = b. The base 'a' must be positive and not equal to 1.

Key Logarithmic Properties: Your Toolkit for Success

Several properties of logarithms are crucial for solving exponential equations. These properties enable us to manipulate and simplify logarithmic expressions:

1. Product Rule: log_a(xy) = log_ax + log_ay

2. Quotient Rule: log_a(x/y) = log_ax - log_ay

3. Power Rule: log_axⁿ = n log_ax

4. Change of Base Rule: log_ax = (log_bx) / (log_ba) This is particularly useful when dealing with logarithms of different bases.

5. Logarithm of 1: log_a1 = 0 (for any base a > 0 and a ≠ 1)

6. Logarithm of the base: log_aa = 1

Solving Basic Exponential Equations using Logarithms

Let's start with some fundamental examples demonstrating how to apply logarithms to solve exponential equations.

Example 1: Solve 2^x = 8

Solution:

1. Take the logarithm of both sides: log(2^x) = log(8)

2. Apply the power rule: x log(2) = log(8)

3. Solve for x: x = log(8) / log(2)

4. Simplify (using a calculator or logarithmic tables): x = 3

Therefore, the solution to 2^x = 8 is x = 3.

Example 2: Solve 5^x = 125

Solution:

1. Take the logarithm of both sides: log(5^x) = log(125)

2. Apply the power rule: x log(5) = log(125)

3. Solve for x: x = log(125) / log(5)

4. Simplify: x = 3

Example 3: Solve 3^x = 10

Solution:

1. Take the logarithm of both sides: log(3^x) = log(10)

2. Apply the power rule: x log(3) = log(10)

3. Solve for x: x = log(10) / log(3)

4. Simplify (using a calculator): x ≈ 2.096

Solving More Complex Exponential Equations

Let's tackle more challenging exponential equations involving multiple terms or different bases.

Example 4: Solve 2^x + 5 = 13

Solution:

1. Isolate the exponential term: 2^x = 8

2. Solve the resulting basic equation (as shown in Example 1): x = 3

Example 5: Solve e^2x = 7

Solution:

1. Take the natural logarithm (ln) of both sides (since the base is e): ln(e^2x) = ln(7)

2. Simplify (ln(e^2x) = 2x): 2x = ln(7)

3. Solve for x: x = ln(7) / 2

4. Simplify (using a calculator): x ≈ 0.973

Example 6: Solve 2^x = 3^x-1

Solution: This requires taking logarithms of both sides and utilizing logarithmic properties.

1. Take the logarithm of both sides: log(2^x) = log(3^x-1)

2. Apply the power rule: x log(2) = (x-1) log(3)

3. Expand: x log(2) = x log(3) - log(3)

4. Rearrange to solve for x: x log(2) - x log(3) = -log(3)

5. Factor out x: x (log(2) - log(3)) = -log(3)

6. Solve for x: x = -log(3) / (log(2) - log(3))

7. Simplify (using a calculator): x ≈ 2.71

Handling Exponential Equations with Multiple Exponential Terms

Some exponential equations have multiple terms with exponents. Solving these requires careful manipulation and potentially the use of substitution.

Example 7: Solve 2^2x - 3(2^x) + 2 = 0

Solution: This equation is quadratic in form. We can use substitution to simplify. Let y = 2^x.

1. Substitute: y² - 3y + 2 = 0

2. Factor the quadratic: (y - 1)(y - 2) = 0

3. Solve for y: y = 1 or y = 2

4. Substitute back: 2^x = 1 or 2^x = 2

5. Solve for x: x = 0 or x = 1

Dealing with Equations Involving Different Bases

Equations with exponential terms of different bases require the use of logarithms to solve. The change of base rule can be particularly useful in these situations.

Example 8: Solve 2^x = 5^x+1

Solution:

1. Take the logarithm (any base) of both sides: log(2^x) = log(5^x+1)

2. Apply the power rule: x log(2) = (x+1) log(5)

3. Expand: x log(2) = x log(5) + log(5)

4. Rearrange: x log(2) - x log(5) = log(5)

5. Factor: x (log(2) - log(5)) = log(5)

6. Solve for x: x = log(5) / (log(2) - log(5))

7. Simplify (using a calculator): x ≈ -1.757

Applications of Solving Exponential Equations

Solving exponential equations is vital in numerous fields, including:

• Population Growth/Decay: Modeling population growth or radioactive decay.
• Compound Interest: Calculating the future value of investments.
• Cooling/Heating: Determining the temperature of an object over time.
• Chemical Reactions: Modeling reaction rates.
• Physics: Analyzing phenomena involving exponential relationships.

Conclusion

Solving exponential equations with logarithms requires a strong understanding of logarithmic properties and algebraic manipulation. By mastering these techniques, you can effectively tackle a wide range of exponential equations, from simple to complex. Remember to carefully follow the steps, apply the appropriate logarithmic properties, and check your answers when possible. Practice is key to building proficiency in solving exponential equations and unlocking their numerous applications in various fields. Consistent practice with different types of problems will solidify your understanding and make you confident in solving even the most challenging exponential equations. Remember to utilize online calculators or logarithmic tables when necessary to assist with calculations.