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Solve The Following Exponential Equation Calculator

Reviewed by Calculator Editorial Team

Exponential equations appear in many scientific and mathematical applications. This calculator helps you solve equations of the form ax = b, where a and b are positive real numbers and a ≠ 1. We'll cover both natural logarithms (ln) and common logarithms (log).

Introduction

Exponential equations are equations where the variable appears in the exponent. The general form is:

ax = b

Where:

  • a is the base (a > 0, a ≠ 1)
  • x is the exponent (the variable we're solving for)
  • b is the result (b > 0)

To solve for x, we use logarithms. There are two common types of logarithms:

  • Natural logarithm (ln) with base e ≈ 2.71828
  • Common logarithm (log) with base 10

How to Solve Exponential Equations

Step 1: Take the logarithm of both sides

Apply the same logarithm to both sides of the equation. This can be either natural logarithm (ln) or common logarithm (log).

logc(ax) = logc(b)

Step 2: Apply logarithm power rule

Use the logarithm power rule which states that logc(ax) = x·logc(a).

x·logc(a) = logc(b)

Step 3: Solve for x

Divide both sides by logc(a) to isolate x.

x = logc(b) / logc(a)

Note: This formula works for any logarithm base c. For natural logarithms, use base e (ln). For common logarithms, use base 10 (log).

Worked Examples

Example 1: Solve 2x = 8 using natural logarithms

Step 1: Take the natural logarithm of both sides

ln(2x) = ln(8)

Step 2: Apply the logarithm power rule

x·ln(2) = ln(8)

Step 3: Solve for x

x = ln(8) / ln(2) ≈ 3

Verification: 23 = 8, which matches the original equation.

Example 2: Solve 10x = 100 using common logarithms

Step 1: Take the common logarithm of both sides

log(10x) = log(100)

Step 2: Apply the logarithm power rule

x·log(10) = log(100)

Step 3: Solve for x

x = log(100) / log(10) = 2

Verification: 102 = 100, which matches the original equation.

Common Mistakes

  • Forgetting to take the logarithm of both sides - This is a critical step that must be done first.
  • Incorrectly applying the logarithm power rule - Remember that logc(ax) = x·logc(a), not ax·logc(a).
  • Using the wrong logarithm base - Make sure to use the same logarithm base on both sides of the equation.
  • Dividing by zero - This occurs when a = 1, which is not allowed in exponential equations.
  • Forgetting to verify the solution - Always plug your solution back into the original equation to ensure it's correct.

Frequently Asked Questions

What is the difference between natural and common logarithms?

Natural logarithms (ln) use base e (approximately 2.71828), while common logarithms (log) use base 10. The choice between them depends on the context and the preferred base for calculations.

Can I solve exponential equations with any logarithm base?

Yes, the formula x = logc(b) / logc(a) works for any logarithm base c. However, natural logarithms (base e) and common logarithms (base 10) are most commonly used in practice.

What if the base a is between 0 and 1?

When the base a is between 0 and 1, the solution x will be negative. This is because the exponential function decreases as x increases when 0 < a < 1.

How do I know if my solution is correct?

Always verify your solution by plugging it back into the original equation. For example, if you solved 2x = 8 and got x = 3, verify by calculating 23 = 8.