Answer:
To solve the equation 5^(2x) * 20^x = 20, we can simplify it by breaking down the terms and applying the properties of exponents.
1. Rewrite 20 as 5 * 4 since 20 can be factored into 5 and 4.
2. Apply the property of exponents: a^(m * n) = (a^m)^n.
Rewrite 5^(2x) as (5^2)^x = 25^x.
Rewrite 20^x as (5 * 4)^x = 5^x * 4^x.
3. Substitute these values back into the equation: 25^x * 5^x * 4^x = 20.
4. Combine like terms by adding the exponents with the same base.
The exponents of 25^x, 5^x, and 4^x can be added together: x + x + x = 3x.
Now we have the equation 25^x * 5^x * 4^x = 20 can be rewritten as 25^3x = 20.
From here, we need to solve for x. Let's take the logarithm (base 25) of both sides of the equation to eliminate the exponent:
log base 25 of (25^3x) = log base 25 of 20.
Applying the logarithmic property, log base a of (a^b) = b, the equation becomes:
3x = log base 25 of 20.
Finally, solve for x by dividing both sides of the equation by 3:
x = (log base 25 of 20) / 3.
So the solution for x is (log base 25 of 20) divided by 3.
Explanation: