Answered

Match each logarithm to its value.


A$\log _{27} 3$
B$\log _{81} 27$
C$\log _9 27$


  1. Logarithm □ is equal to -3.
  2. Logarithm □ is equal to $\frac{3}{2}$.
  3. Logarithm □ is equal to $\frac{3}{4}$.
  4. Logarithm □ is equal to $\frac{1}{3}$.

Verified Answer :

  • Simplify each logarithmic expression by rewriting the base and argument as powers of 3: \(\log_{3^n} (3^m) = \frac{m}{n}\).
  • Evaluate Expression A: \(\log_{27} 3 = \log_{3^3} 3^1 = \frac{1}{3}\).
  • Evaluate Expression B: \(\log_{81} 27 = \log_{3^4} 3^3 = \frac{3}{4}\).
  • Evaluate Expression C: \(\log_{9} 27 = \log_{3^2} 3^3 = \frac{3}{2}\).
  • Final matches: Logarithm A is \(\frac{1}{3}\), Logarithm B is \(\frac{3}{4}\), Logarithm C is \(\frac{3}{2}\). The value \(-3\) remains unmatched.

Explanation

  1. Understanding Logarithmic Properties

To solve the problem of matching each given logarithm to its numerical value, we first simplify each expression by expressing the bases and arguments as powers of a common prime number. Specifically, for any logarithm \(\log_{b} x\), if we can write \(b = a^n\) and \(x = a^m\), then the logarithm is given by:
\[\log_{a^n} (a^m) = \frac{m}{n}\]
where \(a\) is a positive number other than 1. This property arises from the definition \(\log_{b} x = y \iff b^y = x\), which implies \((a^n)^y = a^{ny} = a^m\), hence \(ny = m\) or \(y = \frac{m}{n}\).

  1. Evaluating Expression A

For Expression A, we have \(\log_{27} 3\). We observe that both 27 and 3 can be expressed as powers of 3:
\[27 = 3^3\]
\[3 = 3^1\]
Thus, the expression becomes:
\[\log_{27} 3 = \log_{3^3} (3^1) = \frac{1}{3}\]
This matches one of our target values.

  1. Evaluating Expression B

For Expression B, we have \(\log_{81} 27\). We express both 81 and 27 as powers of 3:
\[81 = 3^4\]
\[27 = 3^3\]
Thus, the expression becomes:
\[\log_{81} 27 = \log_{3^4} (3^3) = \frac{3}{4}\]
This matches one of our target values.

  1. Evaluating Expression C

For Expression C, we have \(\log_{9} 27\). We express both 9 and 27 as powers of 3:
\[9 = 3^2\]
\[27 = 3^3\]
Thus, the expression becomes:
\[\log_{9} 27 = \log_{3^2} (3^3) = \frac{3}{2}\]
This matches one of our target values.

  1. Conclusion and Mapping

Comparing our results:

  • Expression A (\(\log_{27} 3\)) is \(\frac{1}{3}\).
  • Expression B (\(\log_{81} 27\)) is \(\frac{3}{4}\).
  • Expression C (\(\log_{9} 27\)) is \(\frac{3}{2}\).

The target values provided were \(-3, \frac{3}{2}, \frac{3}{4}, \frac{1}{3}\). We have successfully matched A, B, and C to their respective values, leaving \(-3\) as the unused value.

Examples

Logarithms are essential in measuring scales where values span several orders of magnitude. For instance, the Richter scale for earthquake magnitude or the decibel scale for sound intensity uses logarithms to make large, unwieldy numbers manageable. By understanding these properties, you can effectively compare intensities or levels across different media, which is fundamental in fields ranging from sound engineering to geophysics. Just as we converted these logs to a common base to compare them, professionals convert data into logarithmic scales to make comparisons easier.

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