Final answer:
To solve the equation log₄(y-9) + log₄ 3 = log₄ 81, we combine the logarithms on the left side using the product rule, set the inside of the log equal to 81, and then solve for y to find that y equals 36.
Step-by-step explanation:
The question involves solving an equation with logarithms. The equation given is log₄ (y-9) + log₄ 3 = log₄ 81. We can use the properties of logarithms to simplify and solve this equation. First, recall that adding two logs with the same base is the same as taking the log of the product of their arguments, according to the property log a + log b = log(ab). Therefore, the left side of the equation simplifies to log₄ ((y-9) × 3).
Next, since the logarithms on both sides have the same base, their arguments must be equal. Hence, we can set the argument in the single logarithm on the left side equal to the argument of the logarithm on the right side, which gives us the equation (y-9) × 3 = 81. Simplifying further, we get (y-9) = 27, and thus, y = 36.
It is also important to note that the logarithm of a number less than one is negative, and that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. However, these principles are not directly applicable to solving the given question.