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.