What Are Exponential Equations?

Understanding Exponential Equations is crucial for students preparing for board exams, competitive tests like JEE, or anyone who wants to grasp patterns of rapid growth and decay in real life. Mastery of exponential equations helps in algebra, compound interest, and science, making it a key concept for exams and practical situations.

What are Exponential Equations?

An exponential equation is an equation where the variable appears as an exponent. In other words, the unknown value is found in the power position, such as \( a^{x} = b \). Solving these equations often involves rewriting bases, using properties of exponents, or applying logarithms if the bases are different. Exponential equations are commonly used to model real-world phenomena like population growth, radioactive decay, and calculating compound interest.

For example, the equation \( 2^x = 16 \) is an exponential equation because the variable \( x \) is in the exponent.

Important Formulas and Methods

Here are some key formulas and methods to solve exponential equations:

Step-by-Step Worked Examples

Example 1: Solving with Same Base

Solve \( 4^{x} = 16 \).

1. Rewrite 16 as a power of 4: \( 16 = 4^2 \).
2. The equation becomes \( 4^x = 4^2 \).
3. Since the bases are equal, set exponents equal: \( x = 2 \).

Example 2: Different Bases, Use Logarithms

Solve \( 2^x = 20 \).

1. Bases cannot be made equal, so use logarithms: Take log both sides.
2. \( \log(2^x) = \log(20) \)
3. By properties of logs: \( x \cdot \log 2 = \log 20 \)
4. \( x = \dfrac{\log 20}{\log 2} \approx \dfrac{1.3010}{0.3010} \approx 4.32 \)

Example 3: Real-World (Compound Interest)

A sum of ₹10,000 is invested at 10% annual compound interest. After how many years will it double?

1. Formula: \( A = P(1 + r)^t \), \( A = 2P \), \( r = 0.10 \), \( P = 10,000 \).
2. \( 2 = (1.10)^t \)
3. Take log both sides: \( \log 2 = t \cdot \log 1.10 \)
4. \( t = \dfrac{\log 2}{\log 1.10} \approx \dfrac{0.3010}{0.0414} \approx 7.27 \) years

Practice Problems

• Solve for \( x \): \( 5^{x} = 125 \)
• Solve for \( x \): \( 2^{2x+1} = 32 \)
• Solve for \( y \): \( 3^{y-2} = 1/9 \)
• Solve for \( x \): \( 4^{x} = 7 \) (Give answer in logarithmic form)
• The bacteria count doubles every 3 hours. Write an exponential equation for the count after \( t \) hours if the starting number is 200.

Common Mistakes to Avoid

• Not expressing numbers with a common base when possible (example: writing 16 as \( 2^4 \) instead of \( 4^2 \) when the base is 4).
• Forgetting to apply logarithms correctly when bases are different—always use properties like \( \log a^{x} = x \log a \).
• Mixing up exponent rules, like product or quotient rules (Laws of Exponents).
• Not recognizing that \( 1 = a^0 \) for any nonzero \( a \).
• Careless calculation of logarithmic values—use accurate values for exam questions.

Real-World Applications

Exponential equations are used in a variety of real-world scenarios:

• Calculating compound interest in banking and investments
• Modeling population growth and decay in biology (Exponential Growth, Exponential Distribution)
• Describing radioactive decay in chemistry and physics
• Measuring sound levels and earthquake magnitude (decibel and Richter scales use logarithmic and exponential equations)

At Vedantu, we simplify complex topics like exponential equations so students can confidently solve such real-life and exam problems.

In this topic, you learned what exponential equations are, how to solve them, common pitfalls, and how they apply in daily situations. Consistent practice of these equations will boost your problem-solving skills in school and competitive exams. For deeper learning, explore other related concepts such as logarithmic functions, exponents, and exponential functions on Vedantu.