given log 2 base 7 is 0,3562 and log 3 base 7 is 0,5646 evaluate log 42 base 7 without using calculator or tables
42 = 2*21 = 2 * 3 * 7 (lucky :)
so
log 42 = log 2 + log 3 + log 7
all base 7
but what is log 7 base 7
7^log 7 = 7 but that is 7^1
so log 7 = 1 if base 7
so
log 42 = log 2 + log 3 + 1
no way I will add those numbers without my machine :)
To evaluate log base 7 of 42 without using a calculator or tables, we can use logarithmic properties to rewrite 42 in terms of the given logs.
Firstly, we know that log base 7 of a product is equal to the sum of the logs of the individual numbers. Therefore, we can rewrite 42 as a product of prime factors:
42 = 2 * 3 * 7.
Now, using the property mentioned above, we can express log base 7 of 42 as:
log base 7 (42) = log base 7 (2 * 3 * 7).
Next, we can substitute the given log values:
log base 7 (42) = log base 7 (2) + log base 7 (3) + log base 7 (7).
Since log base 7 of 2 is given as 0.3562 and log base 7 of 3 is given as 0.5646, we can substitute these values:
log base 7 (42) ≈ 0.3562 + 0.5646 + log base 7 (7).
Finally, since log base 7 of 7 is 1, we can substitute this value:
log base 7 (42) ≈ 0.3562 + 0.5646 + 1.
By adding these values together, we get:
log base 7 (42) ≈ 1.9208.
Therefore, log base 7 of 42 is approximately 1.9208.
To evaluate log 42 base 7 without using a calculator or tables, we can use logarithmic properties.
We know that log a (m * n) = log a (m) + log a (n).
Therefore, we can express 42 as a product of factors that are powers of 7.
First, let's express 42 as a product of its prime factors:
42 = 2 * 3 * 7
Now, we can write:
log 42 base 7 = log (2 * 3 * 7) base 7
Using the logarithmic property mentioned above, we can split this into three separate logarithmic expressions:
log 42 base 7 = log 2 base 7 + log 3 base 7 + log 7 base 7
We are given that log 2 base 7 is 0.3562 and log 3 base 7 is 0.5646.
So, we can substitute these values into our equation:
log 42 base 7 = 0.3562 + 0.5646 + log 7 base 7
Since log 7 base 7 is equal to 1 (logarithm of the base to itself), we have:
log 42 base 7 = 0.3562 + 0.5646 + 1
Now, we can simplify the equation:
log 42 base 7 = 1.9208
Therefore, the value of log 42 base 7 is approximately 1.9208.