if log base 10 of 2 = .301
then what is log base of 10 of 2000?
I know how to solve with a calculator but I'm having a hard time doing it without one.
log 2000 = log 2 + log 1000 = 3.301
Ok thanks
pls i need th answer to log6 base to10
To find the value of log base 10 of 2000 without using a calculator, you can utilize the logarithmic properties and the value of log base 10 of 2.
Step 1:
Start by breaking down 2000 into its prime factors. 2000 = 2 * 2 * 2 * 2 * 5 * 5 * 5.
Step 2:
Using the logarithmic properties, we can rewrite log base 10 of 2000 as the sum of the logarithms of its prime factors:
log base 10 of 2000 = log base 10 of (2 * 2 * 2 * 2 * 5 * 5 * 5)
= log base 10 of 2 + log base 10 of 2 + log base 10 of 2 + log base 10 of 2 + log base 10 of 5 + log base 10 of 5 + log base 10 of 5
Step 3:
Since we know that log base 10 of 2 is approximately 0.301, we substitute that value into the equation:
log base 10 of 2000 ≈ 0.301 + 0.301 + 0.301 + 0.301 + log base 10 of 5 + log base 10 of 5 + log base 10 of 5
Step 4:
Now we need to determine the value of log base 10 of 5. We can use the same approach as before, utilizing the value of log base 10 of 2:
log base 10 of 5 ≈ log base 10 of (2 * 2.5)
≈ log base 10 of 2 + log base 10 of 2.5
≈ 0.301 + log base 10 of (2.5)
Step 5:
To find the value of log base 10 of 2.5, we need to use additional mathematical techniques. One common approach is to use logarithmic identities to express 2.5 as a power of 10:
2.5 = 10^(log base 10 of 2.5)
Therefore, log base 10 of 2.5 is equivalent to 1 plus the exponent of 10 required to obtain 2.5:
log base 10 of 2.5 = 1 + log base 10 of 2.5 in exponential form
Step 6:
We can now substitute the value of log base 10 of 2.5 back into step 4:
log base 10 of 5 ≈ 0.301 + (1 + log base 10 of 2.5)
Step 7:
Now we can substitute the value of log base 10 of 5 back into step 3:
log base 10 of 2000 ≈ 0.301 + 0.301 + 0.301 + 0.301 + (0.301 + 1 + log base 10 of 2.5) + (0.301 + 1 + log base 10 of 2.5) + (0.301 + 1 + log base 10 of 2.5)
Step 8:
Simplify the expression:
log base 10 of 2000 ≈ 0.301 * 4 + 1 * 3 + log base 10 of 2.5 * 3
Step 9:
Calculate the expression:
log base 10 of 2000 ≈ 1.204 + 3 + log base 10 of 2.5 * 3
Step 10:
Finally, calculate the value of log base 10 of 2.5 * 3:
log base 10 of 2.5 * 3 = log base 10 of (2.5 * 3)
= log base 10 of 7.5
Using the same logarithmic approach as before, you can find the value of log base 10 of 7.5 and substitute it back into step 9 to obtain the final result.
Please note that this method provides an approximation of log base 10 of 2000, and using a calculator will give you a more precise and accurate answer.