Solving $2^x \cdot 5^y = 0,128$

Question

$x,y \in \mathbb Z$

$2^x \cdot 5^y = 0,128$

$x+y = ?$

My attempt:

I know that

$$0,128 = \frac{128}{1000}$$

$$5^3 = 125$$

$$2^{-3} = \frac{1}{8}$$

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EDIT:

$2^7 = 128$

Then we need to get

$0,128$

Answer 1

Hint: $0.128=2^7\times 10^{-3}=2^4\times 5^{-3}$

Answer 2

You know that $128=2^7$, and $1000=10^3=(2\cdot 5)^3=2^3\cdot 5^3$. Therefore

$$0.128=\frac{128}{1000}=\frac{2^7}{2^3\cdot 5^3}$$

Can you do the rest?

Answer 3

You can write 5 in terms of 10 and 2... so you can write the problem as: $2^x \cdot (\frac{10}{2})^y=.128$

$2^{x-y} \cdot 10^y=.128$

Find $y$ first and then $x $