To simplify the expression [tex]\(\log_3\left(\frac{1}{729}\right)\)[/tex] without using a calculator, follow these steps:
1. Rewrite the Fractional Exponent:
The number 729 can be expressed as a power of 3 because [tex]\(3^6 = 729\)[/tex]. Therefore, [tex]\(\frac{1}{729}\)[/tex] can be written as [tex]\(3^{-6}\)[/tex].
2. Use the Logarithm Property:
Recall the property of logarithms that states [tex]\(\log_b(b^a) = a\)[/tex]. This property asserts that the logarithm of a base raised to an exponent is simply the exponent.
3. Apply the Logarithm Property:
Apply this property to our expression [tex]\(\log_3(3^{-6})\)[/tex]. Here, the base [tex]\(b\)[/tex] is 3 and the exponent [tex]\(a\)[/tex] is -6.
[tex]\[
\log_3(3^{-6}) = -6
\][/tex]
Thus, the simplified form of [tex]\(\log_3\left(\frac{1}{729}\right)\)[/tex] is:
[tex]\[
\boxed{-6}
\][/tex]
