Alan

I assume this is meant to be   \(343^{x+1}=49^\frac{1}{3}\)

This can be written as   \(7^{3(x+1)}=7^\frac{2}{3}\)   because 343 = 7^3 and 49 = 7^2

Hence  3(x+1) = 2/3

3x + 3 = 2/3

x+1 = 2/9

x = 2/9 - 1

x = -7/9

.

The calculator here doesn't generate random numbers. Try the online random number generator at http://www.mathgoodies.com/calculators/random_no_custom.html

.

Speed is distance divided by time.  Use the calculator on the home page here. You should specify the units of time.

Then you can home in further:

00062 → 6.2*10^(-4)

If it's just the real zero you are interested in, start by plotting a graph:

Number of digits given by floor(log₁₀(3^1000)) + 1. or floor(1000*log₁₀(3)) + 1 → 478

If the 29, 719 etc were meant to be half-lives then the original equations are indeed incorrect (they are missing the ln(2)).  However, I don't see that they are explicitly called half-lives anywhere.

If the half life is specified then the decay constant will be (mathematically) irrational because of the ln(2). However, there is no compulsion to specify the half-life; one may specify the decay constant instead.

For real-world situations, both the half-life and the decay constant will only be approximate anyway (because based on experimental measurement with attendant inaccuracies).

I guess the examples provided by the questioner here are artificial examples dreamt up by a teacher.  

I assume this means f(1/g(x)) → 4*(1/3x) - 5 → 4/(3x) - 5

.