$\log _2 5+\log _2 8$
We aim to evaluate the expression \(\log_2 5 + \log_2 8\). The base of the logarithms is 2, and the arguments are 5 and 8. To solve this, we can utilize fundamental logarithmic properties. Specifically, we have the product rule for logarithms, which states that \(\log_b x + \log_b y = \log_b(xy)\). Alternatively, we can evaluate each term separately if one is a simpler power of the base.
Notice that the second term, \(\log_2 8\), can be easily simplified because 8 is a power of the base 2. Since \(8 = 2^3\), we can apply the property \(\log_b(b^n) = n\). Therefore, we have:
\[\log_2 8 = \log_2(2^3) = 3.\]
Now, substituting this value back into the original expression gives:
\[\log_2 5 + 3.\]
This can also be written as \(3 + \log_2 5\).
Alternatively, we can use the product rule \(\log_b x + \log_b y = \log_b(xy)\):
\[\log_2 5 + \log_2 8 = \log_2(5 \times 8) = \log_2 40.\]
Both expressions, \(\log_2 40\) and \(3 + \log_2 5\), are equivalent. In standard mathematical notation, the form \(3 + \log_2 5\) is often considered simplified, while \(\log_2 40\) is a single compact logarithmic term. We conclude that the value is \(\boxed{3 + \log_2 5}\).
Logarithms are incredibly useful in fields like computer science for calculating the complexity of algorithms. For example, if you have a dataset and keep splitting it in half to find a specific value, the number of steps required relates to a logarithm with base 2. If you have 8 items, it takes \(\log_2 8 = 3\) splits to narrow it down. Understanding these properties helps in efficiently managing data and understanding growth rates in various technologies.