Final answer:
The value of log3(16/7) is calculated using the quotient and power rules of logarithms, and with the given values log3(4) and log3(7), it is approximately 0.753.
Step-by-step explanation:
The value of log3(16/7) can be calculated using the properties of logarithms, specifically the quotient rule, which states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. The known values given are log3(4) ≈ 1.262 and log3(7) ≈ 1.771.
We can use these values to find log3(16/7) by expressing 16 as 4².
First, we express 16 as 4² and then use the rules of logarithms to write:
log3(16/7) = log3(4²/7)
Now, applying the quotient rule and the power rule of logarithms (loga(b^c) = c * loga(b)), we get:
log3(4²/7) = log3(4²) - log3(7)
log3(4²/7) = 2 * log3(4) - log3(7)
Filling in the given values, we get:
log3(4²/7) = 2 * 1.262 - 1.771
log3(4²/7) = 2.524 - 1.771
= 0.753
Thus, the value of log3(16/7) is approximately 0.753.