Question

Find log\(_{5}\)(125) + log\(_{2}\)(1/16)

• A. 0
• B. 1
• C. 2
• D. -1

Hint:

Remember that log\(_b\)(1/x) is always the negative of log\(_b\)(x). So log\(_2\)(1/16) = -log\(_2\)(16) = -4.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We need to evaluate the sum of two logarithmic expressions: log\(_{5}\)(125) and log\(_{2}\)(1/16).
Step 2: Key Formula or Approach:
The fundamental definition of a logarithm is log\(_b\)(x) = y if and only if b\(^y\) = x.
We will also use the property log\(_b\)(b\(^n\)) = n.
For the second term, we use the negative exponent rule: b\(^{-n}\) = 1/b\(^n\).
Step 3: Detailed Explanation:
Let's evaluate each term separately.
Term 1: log\(_{5}\)(125)
We need to find the power to which 5 must be raised to get 125.
Since 5\(^3\) = 5 \(\times\) 5 \(\times\) 5 = 125, we have:
\[ \log_{5}(125) = \log_{5}(5^3) = 3 \] Term 2: log\(_{2}\)(1/16)
We need to find the power to which 2 must be raised to get 1/16.
First, express 16 as a power of 2: 16 = 2\(^4\).
So, 1/16 = 1/2\(^4\) = 2\(^{-4}\).
Therefore:
\[ \log_{2}(1/16) = \log_{2}(2^{-4}) = -4 \] Combining the terms:
The original expression is the sum of the two terms:
\[ \log_{5}(125) + \log_{2}(1/16) = 3 + (-4) = -1 \] Step 4: Final Answer:
The value of the expression is -1.