Exponents Calculator

Solve for the exponent n in the equation xⁿ = y. This calculator uses logarithms to find the exponent, showing all steps in the solution process.

Solve for Exponent n

xn = y
Answer: n = 3.0000
This means: 2^3.0000 = 8

Solution Steps

1. Start with the equation:

xn = y

This is our original equation where we need to solve for the exponent n. In this case, we have 2n = 8

2. Take the natural logarithm (ln) of both sides:

ln(xn) = ln(y)

We take ln of both sides because logarithms can help us solve for exponents. The natural logarithm (ln) is especially useful because it's the inverse of e (Euler's number). In our case: ln(2n) = ln(8)

3. Use the logarithm property ln(xn) = n·ln(x):

n·ln(x) = ln(y)

This is a key logarithm property: the ln of a number raised to a power equals the power times the ln of the number. For our values: n·ln(2) = ln(8)

Calculating: n·0.6931 = 2.0794

4. Solve for n by dividing both sides by ln(x):

n = ln(y) ÷ ln(x) = 3.0000

Step by step calculation:

n = ln(8) ÷ ln(2)

n = 2.0794 ÷ 0.6931

n = 3.0000

Verification:

We can verify our answer by plugging it back into the original equation:

23.0000 = 8

2 raised to the power of 3.0000 equals 8

💡 Important Notes

  • The base (x) must be a positive number
  • The result (y) must be a positive number
  • The solution uses natural logarithms to solve for the exponent
  • The result may be a decimal number

🌟 Practical Examples

💰

Investment Growth

If your investment of $10,000 grew to $20,000, and you want to know the annual growth rate:

• Let x = 1 + r (where r is the growth rate)

• y = Final Amount ÷ Initial Amount = $20,000 ÷ $10,000 = 2

• n = number of years

• If n = 5 years, then: x⁵ = 2

• Solving for x: x = 2^(1/5) ≈ 1.15

Therefore, the annual growth rate was approximately 15%

👥

Population Growth

If a city's population doubled from 100,000 to 200,000, find how many years it took at 5% annual growth:

• x = 1.05 (5% growth = 1 + 0.05)

• y = Final Population ÷ Initial Population = 200,000 ÷ 100,000 = 2

• Using our calculator: 1.05ⁿ = 2

• n ≈ 14.2 years

The population took about 14.2 years to double at 5% growth

⚛️

Radioactive Decay

If a radioactive sample decayed to 25% of its original amount, find how many half-lives have passed:

• x = 0.5 (each half-life reduces by half)

• y = 0.25 (25% = 0.25)

• Using our calculator: 0.5ⁿ = 0.25

• n = 2 half-lives

It took 2 half-lives to decay to 25%

🔊

Sound Intensity

If a sound is 1000 times more intense than the reference level, find its decibel level:

• x = 10 (decibels use base 10)

• y = 1000 (intensity ratio)

• Using our calculator: 10ⁿ = 1000

• n = 3

The sound level is 30 decibels (10 × n = 10 × 3 = 30 dB)

🏦

Compound Interest

Find how often interest is compounded if $1,000 grows to $1,100 in a year at 10% annual rate:

• x = (1 + 0.10/n) where n is the number of times compounded

• y = Final Amount ÷ Initial Amount = $1,100 ÷ $1,000 = 1.1

• Using the compound interest formula: x^n = 1.1

• This helps determine the optimal compounding frequency

Compare different compounding frequencies to maximize returns

Why These Examples Matter

  • They show how exponential relationships appear in various fields
  • Help in making predictions and understanding growth/decay rates
  • Useful for financial planning and scientific calculations
  • Demonstrate the practical value of solving for exponents