Let's solve this exponential equation step by step. We have 3 to the power x equals 27 times negative one-third times thirteen-fourths, all raised to the fourth power. First, we need to simplify the expression inside the parentheses. Negative one-third times thirteen-fourths equals negative thirteen-twelfths.
Now we raise negative thirteen-twelfths to the fourth power. Since we're raising to an even power, the negative sign disappears, giving us thirteen-twelfths to the fourth power. This equals thirteen to the fourth power divided by twelve to the fourth power. Calculating these values: thirteen to the fourth power equals 28,561, and twelve to the fourth power equals 20,736. So we get the fraction 28,561 over 20,736.
Next, we multiply our result by 27. We can rewrite 27 as 3 to the third power. We also factor 12 to the fourth power as 3 to the fourth times 4 to the fourth. Substituting these, we get 3 cubed times our fraction over 3 to the fourth times 4 to the fourth. Simplifying the powers of 3, we get one-third times the fraction over 4 to the fourth. Since 4 to the fourth equals 256, our final result is 28,561 over 768.
Now we have our simplified equation: 3 to the power x equals 28,561 over 768. To solve for x, we take the logarithm base 3 of both sides. This gives us our final answer: x equals log base 3 of the fraction 28,561 over 768. This is the exact solution to our exponential equation.
Let's summarize our complete solution. We started with the equation 3 to the power x equals 27 times the complex expression raised to the fourth power. Through careful algebraic manipulation, we simplified this to 3 to the power x equals 28,561 over 768. Taking the logarithm base 3 of both sides gives us our final answer: x equals log base 3 of 28,561 over 768. This solution demonstrates the power of systematic algebraic simplification and logarithmic properties in solving exponential equations.