Question

use properties of logarithms to condense the logarithmic expression. write the expression as a single logarithm whose coefficient is 1. where possible, evaluate logarithmic expressions. \\(\log 40 + \log 25\\) \\(\log 40 + \log 25 = \square\\) (simplify your answer.)

Explanation:

Step1: Recall logarithm product rule

The product rule of logarithms states that $\log_a M + \log_a N = \log_a (M \times N)$ for the same base $a$ (assuming the base is the same, usually base 10 or natural log, here we can assume base 10 or the base is consistent). So for $\log 40 + \log 25$, we can apply this rule.

Step2: Apply the product rule

Using the rule, we have $\log 40 + \log 25=\log(40\times25)$. Calculate $40\times25 = 1000$. So now we have $\log(1000)$.

Step3: Simplify $\log(1000)$

We know that $10^3 = 1000$, and if we assume the base of the logarithm is 10 (common logarithm), then $\log_{10} 10^3=3$ (by the property $\log_a a^x = x$).

Answer:

3