Final answer:
To calculate log14 72 given certain logarithm values, we apply the change of base formula and properties of logarithms to express this in terms of known common logarithm values, resulting in approximately 1.6202.
Step-by-step explanation:
To compute log14 72 using the given values for log10 2, log10 3, and log10 7, we need to apply the change of base formula and properties of logarithms. The change of base formula allows us to express a logarithm in terms of logarithms of another base that we know.
First, let's use the change of base formula to express log14 72 in terms of common logarithms (base 10):
log
14
72 = log
10
72 / log
10
14
Break down the numerator and denominator further using the properties of logarithms:
log
10
72 = log
10
(2³ * 3²) = 3 * log
10
2 + 2 * log
10
3
log
10
14 = log
10
(2 * 7) = log
10
2 + log
10
7
Substitute the known values:
log
10
72 = 3 * 0.3010 + 2 * 0.4771
log
10
14 = 0.3010 + 0.8451
Thus:
log
10
72 = 0.9030 + 0.9542 = 1.8572
log
10
14 = 1.1461
Finally, divide the two results to find log14 72:
log
14
72 = 1.8572 / 1.1461 ≈ 1.6202
Therefore, log14 72 is approximately 1.6202.