Convert Powers

Converting the base of powers, calculating the resulting exponent and vice versa. The base of a power indicates how quickly the value increases with each step. With a base of two, the value doubles when the exponent is increased by 1. With a base of ten, it increases tenfold. Functions of the form a^x with parameter a and variable x are called exponential functions. These are characterized by extremely rapid growth and are needed for many practical and theoretical applications in mathematics. We use powers of ten, for example, to represent numbers in the decimal system.

Here, individual powers of the form c * a^x can be converted to those of the form d * b^y. The multipliers c and d are set to the default value of 1. The base a and the exponent x must be entered, and either b or y to calculate the exponent y or the base b, respectively. The base for a and b can also be e for Euler's number, or expressions like 2*e (with *, not 2e). These are converted internally into numerical values. b and y are output as decimal fractions.

	Value 1
Parameter c:
Base a:
Exponent x:
	Value 2
Parameter d:
Base b:
Exponent y:

Round to decimal places.

The formulas for the calculation are::
y=ln((c*a^x)/d) / ln(b)
b=((c*a^x)/d)^(1/y)
using the natural logarithm ln and ^ as exponents. Instead of the natural logarithm ln, which is the one to base e, you can also use a logarithm to a different base, as long as it is the same for the numerator and denominator.

An example from physics is radioactive decay, described by the equation N(t) = N₀*e^(-λ*t), where N₀ is the initial quantity, λ is the decay constant, t is the time, and N(t) is the quantity remaining after time t. Assuming N₀ = 1000 particles, λ = 0.1 s^-1, and t = 10 s, then N(t) = 1000*e^(-0.1*10) = 1000*e^(-1) ≈ 367.9. This should be converted into the form d*b^y, for example, with base 10. Given c = 1000, a = e, x = -1 (because x = λ*t is negative), d = 1, and b = 10, we are looking for the exponent y. Using the formula y = ln((c*a^x)/d) / ln(b), we get y = ln(1000*e^-1 / 1) / ln(10) = (ln(1000) - ln(e)) / ln(10) ≈ (6.9078 - 1) / 2.3026 ≈ 2.566. Therefore, N(t) ≈ 1*10^2.566 ≈ 367.9, which exactly matches the original value.
The base e is the mathematically standard and most useful one, but it is not very intuitive, unlike the base 10, which we are more familiar with. This may be one reason for such conversions.
Values for this example: c = 1000, a = e, x = -1, d = 1, b = 10