Solve for real t 11^(t+1) -1=4*11^t

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

11^t =
y

We'll express 11^(t+1)=(11^t)*11, 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.

We'll move all terms to one side and we'll
get:

11*11^t -4*11^t-1 = 0

But
11^t=y:

11y - 4y - 1 = 0

We'll
combine like terms:

7y - 1 =
0

We'll add 1 both sides:

7y =
1

y = 1/7

But 11^t =
y=1/7

11^t = 1/7

We'll take
logarithms both sides:

ln (11^t) = ln
(1/7)

t*ln11 = ln (1/7)

t = ln
(1/7)/ln11

The real solution of the given
equation is t=ln (1/7)/ln11.