Question

Which one of the following statements is true?

• A. Logarithm of 1 to any non-zero base is 0
• B. Logarithm of any non-zero number to the same base is 1
• C. Logarithms of a number with different bases have different values
• D. All of the above

Answer 1

The Correct Option is D

Step 1: Analyze each statement.

(1) Logarithm of 1 to any non-zero base is 0.

The logarithmic identity states that for any base \( b > 0 \) and \( b \neq 1 \):

\[ \log_b(1) = 0. \]

This is because \( b^0 = 1 \). Hence, this statement is true.

(2) Logarithm of any non-zero number to the same base is 1.

The logarithmic identity states that for any base \( b > 0 \) and \( b \neq 1 \):

\[ \log_b(b) = 1. \]

This is because \( b^1 = b \). Hence, this statement is true.

(3) Logarithms of a number with different bases have different values.

The value of a logarithm depends on both the number and the base. For example:

\[ \log_2(8) = 3, \quad \text{but} \quad \log_4(8) = 1.5. \]

Thus, the logarithm of the same number can indeed have different values depending on the base. This statement is true.

(4) All of the above.

Since all three statements (1), (2), and (3) are true, this option is true.

Final Answer: The correct option is \( \mathbf{\text{All of the above}} \), which corresponds to option \( \mathbf{(4)} \).