Question

question
evaluate:
\\(\log_{81} 3\\)
answer attempt 1 out of 2

Explanation:

Step1: Recall the change of base formula

The change of base formula for logarithms is $\log_{a}b = \frac{\log_{c}b}{\log_{c}a}$ (we can use $c = 10$ or $c = e$, here we'll use natural logarithm or common logarithm, but also we can use the property of exponents. We know that $81=3^4$, so we can use the property $\log_{a^n}a^m=\frac{m}{n}$).
Let's use the property of logarithms: if $y = \log_{a^n}a^m$, then by the definition of logarithm, $(a^n)^y=a^m$, which implies $a^{ny}=a^m$, so $ny = m$, so $y=\frac{m}{n}$.
Here, $a = 3$, $n = 4$ (since $81 = 3^4$), and $m = 1$ (since we have $\log_{81}3=\log_{3^4}3^1$).

Step2: Apply the property

Using the property $\log_{a^n}a^m=\frac{m}{n}$, with $a = 3$, $n = 4$, $m = 1$, we get $\log_{3^4}3^1=\frac{1}{4}$.

Answer:

$\frac{1}{4}$