Given that log4 = 0.6021 and log6 = 0.7782, without using mathematical tables or calculate, evaluate log0.096.
you are apparently using logs base 10, so
log 0.096 = log(96*10^-3)
= log96 - 3
= log(16*6)-3
= 2log4 + log6 - 3
...
2.9824
-2.9824
I think the correct answer should be -1.017728..., and not -2.9824
Thank you
No response
Mathematician
To find the value of log0.096 using the given information without using mathematical tables or calculation, we can use logarithmic properties.
First, recall that logarithms are exponents. Specifically, loga(b) represents the power to which a must be raised to obtain b. Using this property, we can rewrite the expression log0.096 as log(96/1000).
Next, utilize another property of logarithms: log(a/b) = log(a) - log(b). We can apply this property to rewrite log(96/1000) as log(96) - log(1000).
Now, our main objective is to find the values of log(96) and log(1000). However, we only have log4 and log6 provided. We need to use the properties of logarithms creatively to determine the answer.
Notice that 96 = 3 * 32, while 1000 = 10 * 100 = 10 * (10 * 10) = 10^3.
Using the property log(ab) = log(a) + log(b), we can rewrite log(96) and log(1000) as log(3) + log(32) and log(10) + log(10) + log(10) = 3log(10) = 3.
Now, recall the logarithmic properties loga(a) = 1 and loga(1) = 0. By using these properties, we can rewrite log(3) as log(3 * 1) = log(3) + log(1) = log(3) + 0 = log(3).
Finally, substitute the known values log4 = 0.6021, log6 = 0.7782, and log(3) = log(96) - log(1000) = 0.6021 + 2 * 0.7782 - 3 = 0.6021 + 1.5564 - 3 = -0.8415.
Therefore, log0.096 is approximately -0.8415.