Given that log 4 = 0.6021 and log 6 = 0. 7782, evaluate log 0.096
these are base ten logs
0.096 = 4 * 4 * 6 / 10^3
log 0.096 = log 4 + log 4 + log 6 - 3 log 10
but log10 is 1
so
2 log 4 + log 6 - 3
Well, I can certainly calculate that for you, but I must warn you that logarithms can be a bit tricky. They can make you feel as lost as a clown in a maze!
Now, to evaluate log 0.096, we can use the logarithmic property that states log(ab) = log(a) + log(b). Since 0.096 is a decimal number, let's rewrite it as 96 * 10^-3.
So, we can rewrite log 0.096 as log(96 * 10^-3).
Now, applying the logarithmic property, we get log(96) + log(10^-3).
Using the given logarithm values for 4 and 6, we can plug them in:
log(96) = log(4 * 24) = log(4) + log(24)
Now, to get the log(24), we can rewrite it as log(6 * 4). So,
log(24) = log(6) + log(4)
Now you just plug in the given logarithm values and add them up:
log(96) = log(4) + (log(6) + log(4))
Now, adding 0.6021 (log 4) and 0.7782 (log 6), we get:
log(96) ≈ 0.6021 + 0.7782 + 0.6021 = 1.9804
So, log 0.096 is approximately 1.9804. But don't worry, I won't leave you feeling as lost as a clown in a maze!
To evaluate log 0.096, we can use the logarithmic property that states log(a/b) = log(a) - log(b).
First, we need to rewrite 0.096 as a fraction. It can be written as 96/1000.
Next, we can rewrite log(0.096) as log(96/1000).
Using the logarithmic property, we can split this into two parts: log(96) - log(1000).
Now, we can calculate each part separately.
Using the given information, log(96) can be calculated by multiplying log(4) by 2, as 96 is 24.
log(96) = 2 * log(4) = 2 * 0.6021 = 1.2042.
Similarly, using the given information, log(1000) can be calculated by multiplying log(10) by 3, as 1000 is 10^3.
log(1000) = 3 * log(10) = 3 * 1 = 3.
Finally, we can substitute the values back into log(96) - log(1000).
log(0.096) = log(96) - log(1000) = 1.2042 - 3 = -1.7958.
Therefore, log 0.096 is approximately equal to -1.7958.
To evaluate log 0.096 using the given logarithmic values, you can use logarithmic properties to simplify the expression.
First, note that 0.096 can be expressed as a fraction: 0.096 = 96/1000 = 12/125.
Next, we can use the property: log (a/b) = log a - log b. Applying this property to 0.096, we get:
log 0.096 = log (12/125) = log 12 - log 125
Now, we need to find the logarithmic value of 12 and 125. Since we are only given the logarithmic values of 4 and 6, we need to express 12 and 125 in terms of these values.
12 can be expressed as 4 * 3, and 125 can be expressed as 5^3.
Therefore, we can rewrite log 0.096 as:
log 0.096 = log (4 * 3) - log (5^3)
Using the logarithmic property log a + log b = log (ab), we further simplify:
log 0.096 = log 4 + log 3 - log 5^3
Now, substituting the given values:
log 0.096 = 0.6021 + log 3 - (0.7782 * 3)
Simplifying:
log 0.096 = 0.6021 + log 3 - 2.3346
Finally, we need the logarithmic value of 3. Since we only know the logarithmic values of 4 and 6, we need to express 3 in terms of these values.
3 can be expressed as 4 * (3/4).
Substituting this into the equation:
log 0.096 = 0.6021 + log (4 * (3/4)) - 2.3346
Using the property log (a/b) = log a - log b:
log 0.096 = 0.6021 + (log 4 + log (3/4)) - 2.3346
Now, we can use the given logarithmic values:
log 0.096 = 0.6021 + (0.6021 + log (3/4)) - 2.3346
Simplifying further:
log 0.096 = 0.6021 + 0.6021 + log (3/4) - 2.3346
log 0.096 = 1.2042 + log (3/4) - 2.3346
Next, we need to express 3/4 in terms of the given logarithmic values.
3/4 can be expressed as (3 * 6)/(4 * 6) = (3/6) * (6/4) = (1/2) * (3/2) = (1 * 3)/(2 * 2).
Substituting this into the equation:
log 0.096 = 1.2042 + log ((1 * 3)/(2 * 2)) - 2.3346
Using the logarithmic property log (a/b) = log a - log b:
log 0.096 = 1.2042 + (log (1 * 3) - log (2 * 2)) - 2.3346
Simplifying:
log 0.096 = 1.2042 + (log 1 + log 3 - (log 2 + log 2)) - 2.3346
Now, we know that log 1 = 0, so:
log 0.096 = 1.2042 + (0 + log 3 - (log 2 + log 2)) - 2.3346
Now, we can use the given logarithmic values:
log 0.096 = 1.2042 + (0 + 0.7782 - (0.3010 + 0.3010)) - 2.3346
Calculating:
log 0.096 = 1.2042 + 0.7782 - 0.3010 - 0.3010 - 2.3346
log 0.096 = 0.5468
Therefore, log 0.096 is approximately 0.5468.