Solve for t? 50x2^0.4t=400x4^-0.1t

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Divide both sides by 50.

2^(0.4t) = 8*4^-0.1t

We know that 4 = 2^2. so, we will have: 2^(0.4t) = 8*2^(-0.2t).

Multiply both sides by 2^(0.2t).

2^(0.6t) = 8

Take the ln of both sides.

ln(2^(0.6t)) = ln8

0.06t(ln2) = ln8

Isolating the t on left side, we will have:

t = ln8/(0.06ln2) = 5.278031643

So, **t = 5.**

This can also be solved using the rules of exponents (indices):

Divide both sides by 50.

2^(0.4t) = 8*4^-0.1t

`2^(0.4t) = 8times 4^(- 0.1t)`

Reduce everything to the same base:

`2^(0.4t)= 2^3 times 2^2^(- 0.1t)`

Apply the laws of exponents (indices) as we have like bases so we ADD the exponents and the bases fall away effectively:

`0.4t=3+2(- 0.1t)`

Simplify:

`0.4t=3- 0.2t`

Bring your like terms (t) to the same side:

`0.4t` `+0.2t = 3`

`0.6t=3`

`t=3/0.6`

**t=5**

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