Given that log200=0.3010 and log1000=0.4771 calculate without using table log0.72.
first, I think you mean to say that
log 2 = 0.3010
log 3 = 0.4771
Now: 72 = 2^3 * 3^2
so
log 0.72 = 3log2 + 2log3 - 2
You are starting off with a wrong premise
log 200 = 2.301, not as given
log 1000 = 3, not as given
If not stated, log x is assumed to have base 10
If you are working with some mysterious unknown base, then
loga 200 = .301
a^.301 = 200
.301 loga = log200
a = 44,118,214.29
check: 44118214.29^.301 = 200
unfortunately your log 1000 = .4771 does not even that base.
Check your question.
To find log0.72 without using a table, you can use logarithmic properties and the values given for log200 and log1000. Here's how:
1. Recall the logarithmic property: log(a * b) = log(a) + log(b).
2. Rewrite 0.72 as a product of powers of 10. Since 0.72 = 72/100, we can express it as 0.72 = 72 * 0.01.
3. Apply the logarithmic property to the expression found in step 2: log(0.72) = log(72) + log(0.01).
4. Simplify the expression further. First, find the logarithm of each individual factor using the values given for log200 and log1000.
- log(72) can be calculated by finding the logarithm of 8 * 9, as 72 = 9 * 8. Using the logarithmic property, we get log(72) = log(8 * 9) = log(8) + log(9).
- Similarly, log(0.01) can be calculated by expressing it as 0.01 = 100 * 10^(-4). Using the logarithmic property, we get log(0.01) = log(100) + log(10^(-4)).
5. Substitute the values of log200, log1000, and the expanded logarithms of log(72) and log(0.01) into the equation.
log(0.72) = log(8) + log(9) + log(100) + log(10^(-4)).
6. Use the given values log200 = 0.3010 and log1000 = 0.4771.
log(0.72) = log(8) + log(9) + log(100) + log(10^(-4))
= 0.9031 + log(9) + log(100) - 4*log(10)
= 0.9031 + log(9) + log(100) - 4.
7. Finally, calculate the logarithms of 9 and 100 using the given values for log200 and log1000 and substitute them into the equation.
log(0.72) = 0.9031 + log(9) + log(100) - 4
= 0.9031 + 2 * log(3) + 2 - 4
= 0.9031 + 2*0.4771 + 2 - 4
= 0.9031 + 0.9542 + 2 - 4
= 3.8573 - 4
= -0.1427.
Hence, log0.72 ≈ -0.1427.
To calculate log0.72 without using a log table, we can use the following logarithmic identity:
log(a/b) = log(a) - log(b)
In this case, we can rewrite 0.72 as 72/100:
log(0.72) = log(72/100) = log(72) - log(100)
Now, let's calculate each of the logarithms separately:
Using the given values:
log(72) = log(8 * 9) = log(8) + log(9)
Since log(8) is not given, we can try to express it in terms of log(2):
log(8) = log(2^3) = 3 * log(2)
Since we are given the logarithm of 200:
log(2) = log(200) - log(100) = 0.3010 - 0.4771
Now we can substitute the value of log(2) back into log(8):
log(8) = 3 * (0.3010 - 0.4771)
Next, we need to calculate log(9):
log(9) = log(3^2) = 2 * log(3)
Since we are not given the logarithm of 3, we can express it in terms of log(10):
log(3) = log(10/3) = log(10) - log(3)
Using the given values:
log(10) = 1 (log(10) is a commonly known value)
Next, we substitute the value of log(10) back into log(3):
log(3) = 1 - log(10/3)
Finally, we substitute the values of log(8) and log(9) back into log(72):
log(72) = 3 * (0.3010 - 0.4771) + 2 * (1 - log(10/3))
Now, we have calculated log(72).
Next, let's calculate log(100):
We are given the value of log(1000), which is 0.4771.
We know that log(100) = log(1000/10) = log(1000) - log(10) = 0.4771 - 1
Finally, we substitute the values of log(72) and log(100) into log(0.72):
log(0.72) = log(72) - log(100)
Now we can substitute the calculated values to get the final result.