To solve this problem, we'll use the change of base formula for logarithms, which states that:
\[ \log_a(b) = \frac{\log_c(b)}{\log_c(a)} \]
for any positive \(c\). In this case, we want to determine which of the given logarithms has the same value as \(\log_{64}(116)\).
First, we change \(\log_{64}(116)\) to the common logarithm (base 10) or natural logarithm (base \(e\)). We'll use base 10 for this example:
\[ \log_{64}(116) = \frac{\log_{10}(116)}{\log_{10}(64)} \]
Next, let's look at each option provided to find if any of these expressions equal \(\frac{\log_{10}(116)}{\log_{10}(64)}\).
- Option A: \( \log_{27}(19) \)
\[ \log_{27}(19) = \frac{\log_{10}(19)}{\log_{10}(27)} \]
- Option B: \( \log_{\frac{1}{9}}( \frac{1}{2}) \)
Using the change of base formula: \[ \log_{\frac{1}{9}}(\frac{1}{2}) = \frac{\log_{10}(\frac{1}{2})}{\log_{10}(\frac{1}{9})} \] Since \(\log_{10}(\frac{1}{9}) = -\log_{10}(9)\), we have: \[ \log_{\frac{1}{9}}(\frac{1}{2}) = \frac{\log_{10}(\frac{1}{2})}{-\log_{10}(9)} \]
- Option C: \( \log_{12}(18) \)
\[ \log_{12}(18) = \frac{\log_{10}(18)}{\log_{10}(12)} \]
- Option D: \( \log_{16}(164) \)
\[ \log_{16}(164) = \frac{\log_{10}(164)}{\log_{10}(16)} \]
- Option E: \( \log_{36}(1216) \)
\[ \log_{36}(1216) = \frac{\log_{10}(1216)}{\log_{10}(36)} \]
Now, we need to find a logarithm in the options that has the same structure as \(\frac{\log_{10}(116)}{\log_{10}(64)}\).
Let's consider breaking down \(\log_{64}(116)\):
Notice that \(64 = 4^3\) and you can determine the relation to 116, meaning it is essential to calculate \(116\) in terms of its logarithmic base.
After going through the calculations for the values of other expressions, if it equals the form \(\frac{\log_{10}(116)}{\log_{10}(64)}\), we will have our answer.
Upon further inspection, based on the options provided, the candidate expression that will keep this value without resulting to additional transformations with other numbers will be Option D: \( \log_{16}(164)\) since they are closer to a possible factorization in logarithmic terms.
To finalize upon calculating, we find that \( \log_{64}(116)\) needs similar logarithmic characteristics linked to \(164\), taking its factor into account.
As a result, the correct logarithm that will be equivalent to \(\log_{64}(116)\) is:
log16(164).
1 answer