What is the real solution of equation 5^(t+1)=4*5^t+1

We'll apply substitution technique to solve the given
exponential equation.

5^t =
y

The next step would be to express 5^(t+1)=(5^t)*5, based
on the property of multiplying 2 exponential functions, having matching bases. The
result of multiplication will be the base raised to the sum of exponents of each
exponential function.

The equation will
become:

5*5^t -4*5^t-1 = 0

But
5^t=y:

5y - 4y - 1 = 0

We'll
combine like terms:

y - 1 =
0

We'll add 1 both sides:

y =
1

But 5^t = y=1

We could write
1=5^0

5^t=5^0

The
real solution of the given equation is t=0.