Logarithms

To solve this problem, we need to determine the value of in the given logarithmic expression.

1. Simplifying the Logarithmic Expression:
The given equation is: Using the property of logarithms , we can simplify each term: This simplifies to: The sum of the first natural numbers is given by: Thus, the equation becomes:

2. Solving for :
Since appears on both sides of the equation, we can cancel out the term from both sides: The logarithmic equation means that , or .

Final Answer:
The correct answer is (D) 2.

To solve this question, we need to analyze the given logarithmic identity:

1. Given Formula:
The logarithmic identity is:

2. Conditions for the Identity to Hold:
For the identity to be valid, the following conditions must hold true: - , (since the logarithm of a non-positive number is not defined). - and , as the base of a logarithm must be positive and not equal to 1.

3. Identifying the Correct Option:
- Option (A) is incorrect because the value of must not be equal to 1.
- Option (B) is incorrect because and should be positive.
- Option (C) is incomplete because the values of and are not defined as positive.
- Option (D) is correct as it specifies that , , and , which is the required condition for the identity to hold.

Final Answer:
Option (D) is correct.

To solve this problem, we need to simplify the given logarithmic equation:

1. Given Equation:

2. Applying Logarithmic Properties:
Using the property , we can rewrite the equation as:

3. Equating the Arguments:
Since the logarithms are equal, their arguments must be equal as well. Therefore:

4. Solving for :
Cross multiplying the equation gives:

Final Answer:
The correct option is (B) .

To solve the problem, we need to find the correct representation for the expression .

1. Understanding Logarithmic Properties:
We can use the property of logarithms which states:
This property implies that , because the base and the logarithm's base are the same.

2. Verifying the Options:
The given equation corresponds exactly to Option (3):

Final Answer:
The correct option is Option C: .

To solve the problem, we need to evaluate the logarithmic expression .

1. Understanding the Logarithmic Identity:
Recall that means . So we want to find the power to which 0.1 must be raised to get 0.01.

2. Converting to Powers of 10:
We know:

So, the expression becomes:

3. Using Change of Base Formula:
We apply the formula:


So:

Final Answer:
The value of is 2 (Option B).

Step 1: Express 625 as a power of 5 We know that Step 2: Use change of base identity Using the identity , we get Step 3: Simplify further So,

The correct option is (B):