You may also useĀ logarithmation operation, hence, taking logarithms both sides, yields:

`ln (4^(5x-3)) = ln 1`

Since `ln 1 = 0` and ln `(4^(5x-3)) = (5x - 3)*ln 4,` yields:

`(5x - 3)*ln 4 = 0`

Since `ln 4 != 0` , hence, only the factor `5x - 3 = 0` , such that:

`5x - 3 = 0 => 5x = 3=> x = 3/5`

**Hence, solving for x the given equation using logarithmation, yields **`x = 3/5.`

You want to know why the solution of the equation 4^(5x-3)=1 is x = 3/5

For this you have to use one important property of exponents.

x^(a - b) = x^a/x^b

If a = b, x^(a - b) = x^0 and x^a/x^a = 1

Therefore the number on the right hand side of the equation can be written as a power of 4 as 4^0.

The equation is now 4^(5x - 3) = 4^0

As the base is equal equate the exponent

5x - 3 = 0

x = 3/5

As proof, the value of 4^(5x - 3) if x = 3/5 is 4^(5*(3/5) - 3) = 4^(3 - 3) = 4^3/4^3 = 1 which is the right hand side.

Let's see how to get this answer. First, we'll re-write the equation, substituting 1 by 4^0, in this way creating matching bases both sides.

4^(5x-3) = 4^0

Since we have matching bases, we can apply one to one property:

5x - 3 = 0

We'll add 3 both sides:

5x - 3 + 3 = 3

5x = 3

We'll divide by 5:

x = 3/5

So, the answer 3/5 is the value of x that makes the identity to be true.

4^(5*3/5-3) = 1

4^(3-3) = 4^0

4^0 = 4^0 true for x = 3/5