Question

which is equivalent to \\(\log_{2} n = 4\\)? \\(n = \log 4 \cdot \log 2\\) \\(\log n = \frac{\log 2}{4}\\) \\(n = \frac{\log 2}{\log 4}\\) \\(\log n = 4 \log 2\\)

Explanation:

Step1: Recall the definition of logarithms

The logarithmic equation \(\log_{a}b = c\) is equivalent to \(a^{c}=b\). So for \(\log_{2}n = 4\), we can rewrite it in exponential form as \(n = 2^{4}\). But we can also use the change - of - base formula. The change - of - base formula for logarithms is \(\log_{a}b=\frac{\log_{c}b}{\log_{c}a}\) (where \(c>0,c
eq1\)). We can also use the property of logarithms: if \(\log_{a}n = k\), then taking the common logarithm (base 10) of both sides, we have \(\log(n)=\log(a^{k})\) (by the property that if \(y = \log_{a}x\), then \(x=a^{y}\), so \(\log(x)=\log(a^{y})\)). And by the power rule of logarithms \(\log(a^{y})=y\log(a)\).

Step2: Apply the power rule of logarithms

Starting with \(\log_{2}n = 4\). If we take the common logarithm (base 10) of both sides, we get \(\log(\log_{2}n)=\log(4)\)? No, that's not the right way. Wait, the correct way is: from \(\log_{2}n = 4\), we know that \(n = 2^{4}\). Now, take the common logarithm of both sides: \(\log(n)=\log(2^{4})\). By the power rule of logarithms \(\log(a^{b})=b\log(a)\), so \(\log(n)=4\log(2)\).

Answer:

\(\log n = 4\log 2\) (the option \(\boldsymbol{\log n = 4\log 2}\))