Given the log2 =0.3010 and log3=0.4771 calculatewithout using mathematical tables or calculator the value of
a.log54
b.log2.4
c.log30
d.log540
a. log54 = log(3^3 * 2) = 3log3 + log2 = 3(0.4771) + 0.3010 = 1.7323
b. log2.4 = log2 + log(1.2) = 0.3010 + log(6/5) [logarithmic identity: loga.b = loga + logb] = 0.3010 + log6 - log5 - log2 [logarithmic identities: loga/b = loga - logb, loga^b = b*loga] = 0.3010 + 1.7782 - 0.6989 - 1 = 0.3803
c. log30 = log(3 * 10) = log3 + log10 = log3 + 1 [logarithmic identity: loga.b = loga + logb] = 0.4771 + 1 = 1.4771
d. log540 = log(3^3 * 2^2 * 5) = 3log3 + 2log2 + log5 = 3(0.4771) + 2(0.3010) + 0.6989 = 2.7323
log 2.4 = log 24 - log 10 = log 24 - 1
= log (2*2*2*3) - 1 = log 2 + log 2 + log 2 + log 3 - 1
That is correct, but note that the value of log24 that you used is not exact (it is approximately 1.3802), so the final answer will have some degree of approximation.
... but we are not given log 5 , just log 2 and log 3
so I used log (10/2) = log 10 - log 2 = 1 - log 2
You're right, my apologies for my oversight. You used the proper method to evaluate log 2.4.
To calculate the values of a. log54, b. log2.4, c. log30, and d. log540 without using mathematical tables or a calculator, we can use the properties of logarithms. The most important properties for this calculation are the power rule and the product rule of logarithms.
a. log54 = log(2^2 × 3^3)
Applying the product rule of logarithms, log(a × b) = log(a) + log(b):
log54 = log(2^2) + log(3^3)
= 2log2 + 3log3
= 2(0.3010) + 3(0.4771)
= 0.6020 + 1.4313
= 2.0333
b. log2.4 = log(2 × 2.4)
Applying the product rule of logarithms:
log2.4 = log(2) + log(2.4)
= 1(log2) + log2.4
= 1(0.3010) + log2.4
= 0.3010 + log2.4
c. log30 = log(2 × 3 × 5)
Applying the product rule of logarithms:
log30 = log(2) + log(3) + log(5)
= log2 + log3 + log5
= 0.3010 + 0.4771 + log5
d. log540 = log(2^2 × 3^3 × 5)
Applying the product rule of logarithms:
log540 = log(2^2) + log(3^3) + log(5)
= 2log2 + 3log3 + log5
= 2(0.3010) + 3(0.4771) + log5
= 0.6020 + 1.4313 + log5
So, to calculate the exact values for log54, log2.4, log30, and log540 without using a calculator, you need to apply the product and power rules repeatedly and use the given values of log2 and log3 to substitute for each logarithm.