Logarithmic Equation

Question

9.2^2t+1=11.3^t. Round the answer to the nearest thousandth. 

Answer 1

9.2^2t+1 = 11.3^t

ln(9.2^2t+1) = ln(11.3^t)

(2t + 1) ln(9.2) = t ln(11.3)

(2t + 1)/t = ln(11.3) / ln(9.2)

2 + 1/t = ln(11.3) / ln(9.2)

1/t = ln(11.3) / ln(9.2) - 2 ≅ -0.907354

t ≅ 1/(-0.907354) ≅ -1.1021

Test the solution.

9.2^2t+1 = 9.2^2(-1.1021)+1 ≅ 0.06909

11.3^t = 11.3^-1.1021 ≅ 0.06909

 

The solution is good.

Answer 2

ln (9.22t+1)=ln (11.3t)

(2t+1)ln (9.2)=(t)ln (11.3)

(2t+1/t) = ln(11.3) / ln(9.2)

2+1/t = ln (11.3)/ ln(9.2)

1/t = ln(11.3)/ ln(9.2)-2

t=1/((ln(11.3)/ln(9.2))-2)

t=1/(1.093-2)

t= 1/(-907)≅-1.1021