Questions: Solve for (x). [ 9^4x = 16^x+10 ] Round your answer to the nearest thousandth. Do not round any intermediate computations. [ x= ]

Solution Steps

We start with the equation:

\[ 9^{4x} = 16^{x+10} \]

Taking the logarithm of both sides gives us:

\[ 4x \cdot \log(9) = (x + 10) \cdot \log(16) \]

Rearranging the equation, we have:

\[ 4x \cdot \log(9) - x \cdot \log(16) = 10 \cdot \log(16) \]

Factoring out \(x\) from the left side results in:

\[ x \cdot (4 \cdot \log(9) - \log(16)) = 10 \cdot \log(16) \]

Now, we can isolate \(x\):

\[ x = \frac{10 \cdot \log(16)}{4 \cdot \log(9) - \log(16)} \]

Substituting the values of \(\log(9) \approx 2.1972\) and \(\log(16) \approx 2.7726\):

\[ x \approx \frac{10 \cdot 2.7726}{4 \cdot 2.1972 - 2.7726} \]

Calculating this gives us:

\[ x \approx 4.6085 \]

Final Answer