Question

solve for y in the equation below. round your answer to the nearest hundredth. do not round any intermediate computations. $3^{-9y}=6$

Explanation:

Step1: Take log of both sides

Take the natural logarithm (ln) of both sides of the equation \(3^{-9y} = 6\). We get \(\ln(3^{-9y})=\ln(6)\).

Step2: Use logarithm power rule

Using the power rule of logarithms \(\ln(a^b)=b\ln(a)\), the left - hand side becomes \(-9y\ln(3)=\ln(6)\).

Step3: Solve for y

First, we can express \(y\) as \(y =-\frac{\ln(6)}{9\ln(3)}\).
Now, calculate the values: \(\ln(6)\approx1.791759\) and \(\ln(3)\approx1.098612\).
Then \(9\ln(3)=9\times1.098612 = 9.887508\).
\(\frac{\ln(6)}{9\ln(3)}=\frac{1.791759}{9.887508}\approx0.1812\).
So \(y = - 0.1812\approx - 0.13\) (rounded to the nearest hundredth).

Answer:

-0.13