2^(x-1)*5^x=200

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вопрос

To solve for x in the equation 2^(x-1)*5^x=200, we can start by simplifying the equation.

First, we can rewrite 200 as 2^3*5^2.

Therefore, the equation becomes:

2^(x-1)5^x = 2^35^2

Next, we can combine the exponents on the left side of the equation using the properties of exponents. This gives us:

2^(x-1+x)5^x = 2^35^2

Simplify the exponents:

2^(2x-1)5^x = 2^35^2

Now, we can rewrite both sides of the equation with the same base of 2:

(2^2)^(2x-1)5^x = 2^35^2

Simplify:

4^(2x-1)5^x = 825

End the process:

4^(2x-1)*5^x = 200

Since 4 = 2^2, we can rewrite 4^(2x-1) as (2^2)^(2x-1), which simplifies to 2^(4x-2).

This gives us:

2^(4x-2)*5^x = 200

Now we have a simpler equation to work with. We can now set the equation equal to 200:

2^(4x-2)*5^x = 200

And solve for x from here.