Question

Assertion: If log (3x – 5) = 1, then x = 2

Reason: If a^m = n, then log_a n = m

Options

• Both A and R are true and R is the correct reason for A.

• Both A and R are true but R is the incorrect reason for A.

• A is true but R is false.

• A is false but R is true.

Answer 1

A is false but R is true.

Explanation:

Assertion: If log (3x – 5) = 1, then x = 2

Analysis of the assertion:

1. Identify the base: When a logarithm is written without an explicit base, it is assumed to be base 10 the common logarithm.

2. Convert to exponential form: Using the rule from the reason, log_a n = m can be converted to a^m = n.

3. Apply the rule: For log₁₀(3x – 5) = 1, we can rewrite it as 10¹ = 3x – 5.

4. Solve for x:

10 = 3x – 5

15 = 3x

x = 5

5. Check for validity: For a logarithm to be defined, its argument must be positive.

We check the solution 3(5) – 5 = 15 – 5 = 10.

Since 10 > 0, the solution is valid.

6. Compare to assertion: The calculated value x = 5 does not match the asserted value x = 2.

The assertion is false.

Reason: If a^m = n, then log_a n = m

Analysis of the reason:

This is the fundamental definition of a logarithm. It is the rule for converting an equation from exponential form to logarithmic form.

The base of the logarithm is the base of the exponential and the logarithm is equal to the exponent.

The reason is a true mathematical statement.

The assertion is false, but the reason is a true statement