Question

What is the value of the logarithmic or exponential expression? Select true or false for each statement.

- 35 log₃2 = 32
1. True
2. False

- 3⁵ log₃2 = 32
1. True
2. False

Answer 1

Both statements provided in the question are false. The first is a simple multiplication of a constant with a logarithm, and the second treats an exponential expression incorrectly according to logarithmic properties. Understanding the properties of logarithms and their relation to exponentials is key in evaluating these statements.

Step-by-step explanation:

The question is asking to find the value of the logarithmic or exponential expression and to determine whether each given statement about the value of the expression is true or false. Let's examine the two expressions given:

1. -35 log₃ 2 is not equal to 32. This is because -35 is simply a coefficient multiplied by the logarithm of 2 to the base 3, which does not simplify to 32. Therefore, statement 1 is False.
2. 3⁵ log₃ 2 = 32 is tested using the property of logarithms that states the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. In mathematical terms, this property is represented as logₙ(aʸ) = b * logₙ(a). To apply this property here: 3⁵ * log₃ 2 becomes 5 * log₃ 3², which simplifies to 5 * 2 = 10 since log₃ 3² means 'to what power do we raise 3 to get 32', and that power is 5. Therefore, statement 2 is False.