Question

7.1 hw - guess the exponent
score: 5.67/12 answered: 6/12
question 7
evaluate the following expression.
\log_{36}(6)

Explanation:

Step1: Recall the logarithm power rule

We know that for a logarithm \(\log_{a^n}(a)\), we can use the change - of - base formula or the property of logarithms related to exponents. First, note that \(36 = 6^2\). So we can rewrite the logarithm \(\log_{36}(6)\) as \(\log_{6^2}(6)\).

Step2: Apply the logarithm power rule

The power rule of logarithms states that \(\log_{a^b}(c)=\frac{1}{b}\log_{a}(c)\). When \(c = a\), \(\log_{a}(a) = 1\). So for \(\log_{6^2}(6)\), using the formula \(\log_{a^b}(x)=\frac{1}{b}\log_{a}(x)\), here \(a = 6\), \(b = 2\) and \(x = 6\). Then \(\log_{6^2}(6)=\frac{1}{2}\log_{6}(6)\).

Step3: Evaluate \(\log_{6}(6)\)

We know that for any positive number \(a
eq1\), \(\log_{a}(a)=1\). So \(\log_{6}(6) = 1\). Then \(\frac{1}{2}\log_{6}(6)=\frac{1}{2}\times1=\frac{1}{2}\).

Answer:

\(\frac{1}{2}\)