The value of $\log_{(-\frac{1}{3})} 81$ is equal to

Concept & Formula To evaluate a logarithm, we rely on its relationship with exponents. The foundational equation converting a logarithm to exponential form is: $$x = \log_a b \implies a^x = b$$ By setting the given logarithmic expression equal to a variable, we can solve the resulting exponential equation by matching bases. Step-by-Step Solution * **Given:** $\log_{(-\frac{1}{3})} 81$ * **Set up the equation:** Let the value of the expression be $x$. $$\log_{(-\frac{1}{3})} 81 = x$$ * **Convert to exponential form:** $$(-\frac{1}{3})^x = 81$$ * **Express both sides with the same base:** We know that $81$ is a power of $3$ ($3^4 = 81$). We can also write $-\frac{1}{3}$ as a negative power. * Notice that $(-\frac{1}{3})^{-4} = (-3)^4 = 81$. * **Match the exponents:** Since $(-\frac{1}{3})^{-4} = 81$, we can directly deduce the value of $x$. $$x = -4$$ Exam Strategy & Shortcut Whenever you have a fraction as a base and an integer as the argument, expect the answer to be negative. Recognizing that $3^4 = 81$ immediately tells you that the exponent must involve a $4$. Since the base is flipped (a fraction $\frac{1}{3}$), the exponent must be negative to flip it back to a whole number $3$. Thus, $-4$ is the logical quick answer. Common Pitfall A common mistake is ignoring the negative sign on the fraction or simply guessing $4$ because $3^4 = 81$. However, $(-\frac{1}{3})^4 = \frac{1}{81}$, not $81$. Always verify the sign of your resulting exponent by plugging it back into the base. Final Answer **Therefore, the correct answer is -4.**