Question

evaluate logarithms (level 2) unique id: 0058
score: 0.75/4 penalty: 0.25 off
question
what is the value of \\(\log_{9} \sqrt5{9}\\)?
answer attempt 2 out of 2

Explanation:

Step1: Rewrite the radical as an exponent

The fifth root of 9 can be written as \(9^{\frac{1}{5}}\), so the expression becomes \(\log_{9} 9^{\frac{1}{5}}\).

Step2: Apply the logarithm power rule

The power rule of logarithms states that \(\log_{a} b^{c}=c\log_{a} b\). Here, \(a = 9\), \(b = 9\), and \(c=\frac{1}{5}\), so we have \(\frac{1}{5}\log_{9} 9\).

Step3: Evaluate \(\log_{9} 9\)

By the definition of a logarithm, \(\log_{a} a = 1\) for any positive \(a
eq1\). So \(\log_{9} 9 = 1\).

Step4: Multiply to get the result

Multiply \(\frac{1}{5}\) by 1, which gives \(\frac{1}{5}\).

Answer:

\(\frac{1}{5}\)