Expand the Logarithmic Expression log base 2 of 7.3

Problem

(log_2)(7.3)

Solution

1. Identify the decimal value inside the logarithm and rewrite it as a fraction to facilitate expansion.

7.3=73/10

2. Substitute the fraction back into the logarithmic expression.

(log_2)(73/10)

3. Apply the quotient rule for logarithms, which states that (log_b)(M/N)=(log_b)(M)−(log_b)(N)

(log_2)(73)−(log_2)(10)

4. Factor the number 10 into its prime factors, 2 and 5, to expand the second term further.

(log_2)(73)−(log_2)(2⋅5)

5. Apply the product rule for logarithms, which states that (log_b)(M*N)=(log_b)(M)+(log_b)(N) Note the subtraction applies to the entire term.

(log_2)(73)−((log_2)(2)+(log_2)(5))

6. Simplify the expression by distributing the negative sign and evaluating (log_2)(2) which equals 1.

(log_2)(73)−(log_2)(5)−1

Final Answer

(log_2)(7.3)=(log_2)(73)−(log_2)(5)−1

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