- Use the product rule for logarithms \(\log_{b}(M) + \log_{b}(N) = \log_{b}(MN)\) to combine the terms into \(\log_{6}(54 \times 24)\).
- Calculate the product of the arguments \(54 \times 24 = 1296\).
- Recognize that \(1296\) is a power of \(6\), specifically \(6^4\).
- Apply the property \(\log_{b}(b^k) = k\) to find the final value of \(\boxed{4}\).
Explanation
- Applying the Product Rule for Logarithms
The goal is to evaluate the expression \(\log_{6}(54) + \log_{6}(24)\) using the properties of logarithms. We recall the product rule for logarithms, which states that for any positive real bases \(b\) (where \(b
eq 1\)) and positive real arguments \(M\) and \(N\), the sum of two logarithms with the same base is equal to the logarithm of the product of their arguments:
\[\log_{b}(M) + \log_{b}(N) = \log_{b}(M \times N)\]
In our case, the base \(b = 6\), \(M = 54\), and \(N = 24\).
- Combining and Calculating the Product
By applying the product rule to the given expression, we combine the two logarithms into one:
\[\log_{6}(54) + \log_{6}(24) = \log_{6}(54 \times 24)\]
Now, we calculate the product of the arguments:
\[54 \times 24 = 1296\]
So the expression simplifies to \(\log_{6}(1296)\).
- Evaluating the Logarithm
To evaluate \(\log_{6}(1296)\), we need to express \(1296\) as a power of \(6\). We check the powers of 6:
\[6^1 = 6\]
\[6^2 = 36\]
\[6^3 = 216\]
\[6^4 = 216 \times 6 = 1296\]
Since \(1296 = 6^4\), we have:
\[\log_{6}(1296) = \log_{6}(6^4)\]
Using the logarithmic property \(\log_{b}(b^k) = k\), we find that:
\[\log_{6}(6^4) = 4\]
Thus, the value of the expression is \(\boxed{4}\).
Examples
Logarithms are used extensively in fields like acoustics (measuring sound intensity in decibels), chemistry (measuring pH levels of substances), and finance (calculating compound interest growth). For instance, if you are measuring the sound intensity of two different sources, you might sum their logarithmic values to determine the combined decibel level. Understanding these basic properties allows you to simplify complex calculations into manageable steps, just as we did by combining the terms and simplifying the base power.