To determine which expression is equivalent to [tex]\(\log_2 h = 4\)[/tex], follow these steps for solving the given equation:
1. Rewrite the logarithmic equation in its exponential form:
[tex]\[
\log_2 h = 4
\][/tex]
This can be rewritten as:
[tex]\[
h = 2^4
\][/tex]
2. Calculate the value of [tex]\(h\)[/tex]:
[tex]\[
2^4 = 16
\][/tex]
Therefore:
[tex]\[
h = 16
\][/tex]
3. Identify which given expression matches the form of [tex]\(\log_2 h = 4\)[/tex]:
We have:
- [tex]\(\log n=\frac{\log 2}{4}\)[/tex]
- [tex]\(n=\frac{\log 2}{\log 4}\)[/tex]
- [tex]\(n=\log 4 \cdot \log 2\)[/tex]
- [tex]\(\log n=4 \log 2\)[/tex]
Since [tex]\(h = 16\)[/tex], note that:
[tex]\[
\log_2 16 = 4
\][/tex]
4. Relate this to the common logarithm (base 10) by using logarithmic properties:
Given [tex]\(\log_2 16 = 4\)[/tex], we can use the change of base formula:
[tex]\[
\log_2 16 = \frac{\log_{10} 16}{\log_{10} 2}
\][/tex]
5. Rewrite the equation [tex]\(\log_2 16 = 4\)[/tex] using common logarithms:
Since [tex]\(16 = 2^4\)[/tex], we have:
[tex]\[
\log_{10} 16 = \log_{10} (2^4) = 4 \cdot \log_{10} 2
\][/tex]
Therefore:
[tex]\[
\log_2 16 = 4 \cdot \log_{10} 2 / \log_{10} 2 = 4
\][/tex]
6. Conclusion:
The expression that matches [tex]\(\log_2 h = 4\)[/tex] can then be written as:
[tex]\[
\log n = 4 \cdot \log 2
\][/tex]
The equivalent expression to [tex]\(\log_2 h = 4\)[/tex] is:
[tex]\[
\boxed{\log n = 4 \log 2}
\][/tex]
