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

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to solve 11^(t+1) -1=4*11^t for t.

11^(t+1) -1=4*11^t

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

let 11^t = x

=> 11x - 1 = 4x

=> 7x = 1

=> x= 1/7

11^t = 1/7

Here the bases are different, so we have to use the log function.

log 11^t = log (1/7)

=> t* log 11 = -log 7

=> t = -log 7 / log 11

The required value of t is -log 7 / log 11

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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.

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