Question

How to solve this? *log*

`(5^(x+1))^2 = 0.2sqrt(5^x)`

Answer 1

`x=-2`

Explanation:

Use the rule `(a^b)^c=a^(bc)` to simplify the left hand side of the equation. Also, write `sqrt(5^x)` as `(5^x)^(1/2)` and simplify it using the same rule, to give:

`5^(2x+2)=0.2(5^(1/2x))`

Now, note that `0.2=1/5=5^-1`.

`5^(2x+2)=5^-1(5^(1/2x))`

Simplify the right hand side using the rule `a^b(a^b)=a^(b+c)`.

`5^(2x+2)=5^(1/2x-1)`

Now, since we have two exponential functions with the same base being equal, we also know that their exponents will be equal.

`2x+2=1/2x-1`

`3/2x=-3`

`x=-3(2/3)=-2`