How can you find log base 3 of 242/5 using the given information without a calculator?
log base 3 of 10 = 2.1
log base 3 of 11 = 2.2
log base 3 of 4 = 1.3
To find log base 3 of 242/5 without a calculator, we can use the properties of logarithms to rewrite the expression in terms of the given information. We will use the fact that log base 3 of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator.
log base 3 of (242/5) = log base 3 of 242 - log base 3 of 5
First, let's find log base 3 of 242 and log base 3 of 5 using the given information.
Since we don't have the exact value for log base 3 of 242 in the given information, we have to find a representation for 242 in terms of numbers whose logarithm we know.
From the given information, we know log base 3 of 10 is 2.1 and log base 3 of 11 is 2.2. We can see that 242 lies between the logarithms of 10 and 11.
To find how many times 10 must be multiplied to result in 242, we can calculate:
(242 / 10) ≈ 24.2
Similarly,
(242 / 11) ≈ 22
So, we can write:
log base 3 of 242 = log base 3 of (10 * 24.2)
= log base 3 of 10 + log base 3 of 24.2
From the given information, we know log base 3 of 10 is 2.1, but we don't have the exact value for log base 3 of 24.2.
Let's find log base 3 of 24.2 using a similar approach as before. We know log base 3 of 4 is 1.3, so 24.2 lies between log base 3 of 4 and log base 3 of 11.
Again, we can find how many times 4 must be multiplied to result in 24.2:
(24.2 / 4) ≈ 6.05
Then, we can write:
log base 3 of 24.2 = log base 3 of (4 * 6.05)
= log base 3 of 4 + log base 3 of 6.05
Since we don't have the exact value for log base 3 of 6.05 in the given information, we need to approximate it. We know that 6.05 is between 4 and 11, so we can approximate:
(6.05 / 4) ≈ 1.5125
Therefore,
log base 3 of 6.05 ≈ log base 3 of (4 * 1.5125)
≈ log base 3 of 4 + log base 3 of 1.5125
Using the given information, we know log base 3 of 4 is 1.3, and we just approximated log base 3 of 1.5125.
Now, we can substitute the values we have found back into the original expression to calculate log base 3 of (242/5):
log base 3 of (242/5) ≈ (2.1 + 1.3) - log base 3 of 5
After calculating the right-hand side of the expression, you should have an approximation for log base 3 of (242/5) without using a calculator.