Question

consider the following equation.
\log_{9} 5^{x + 2} = 16
find the value of x.
round your answer to the nearest thousandth.
\\(x =\\)

Explanation:

Step1: Use the logarithmic identity \( \log_a b^c = c\log_a b \)

We have the equation \( \log_{9} 5^{x + 2}=16 \). By the power rule of logarithms, this can be rewritten as \( (x + 2)\log_{9} 5=16 \).

Step2: Solve for \( x + 2 \)

First, we know that \( \log_{9} 5=\frac{\ln 5}{\ln 9} \) (change of base formula). So we can solve for \( x + 2 \) by dividing both sides of the equation \( (x + 2)\log_{9} 5=16 \) by \( \log_{9} 5 \).
\( x + 2=\frac{16}{\log_{9} 5}=\frac{16}{\frac{\ln 5}{\ln 9}}=\frac{16\ln 9}{\ln 5} \)

Step3: Calculate the value of \( \frac{16\ln 9}{\ln 5} \)

We know that \( \ln 9 = 2\ln 3\approx2\times1.0986 = 2.1972 \) and \( \ln 5\approx1.6094 \)
\( \frac{16\times2.1972}{1.6094}=\frac{35.1552}{1.6094}\approx21.843 \)

Step4: Solve for \( x \)

Now we have \( x + 2\approx21.843 \), so we subtract 2 from both sides to get \( x \):
\( x=21.843 - 2=19.843 \)

Answer:

\( x\approx19.843 \)