knowing that log to the base 2 of 7 is 2.807 what is log to the base 2 of 14
given : log2 7 = 2.807
log2 7
= log2 (2x7)
= log2 2 + log2 7
= 1 + 2.807
= 3.807
To find the value of log to the base 2 of 14, we can use the properties of logarithms. One property states that log(base a)(xy) = log(base a)(x) + log(base a)(y).
Since log(base 2)(7) is given as 2.807, we can rewrite it as:
log(base 2)(7) = log(base 2)(2*3.5) = log(base 2)(2) + log(base 2)(3.5).
Now, we need to find log(base 2)(3.5). Let's continue:
log(base 2)(7) = 1 + log(base 2)(3.5).
To find log(base 2)(14), we can use the property mentioned earlier. Since 14 = 2 * 7, we can write it as:
log(base 2)(14) = log(base 2)(2 * 7) = log(base 2)(2) + log(base 2)(7).
Now we substitute the value of log(base 2)(7) we found earlier:
log(base 2)(14) = log(base 2)(2) + log(base 2)(7) = 1 + log(base 2)(3.5) + log(base 2)(7).
Therefore, log(base 2)(14) = 1 + 2.807 = 3.807
To find log base 2 of 14, we can use the basic property of logarithms:
log base a (mn) = log base a (m) + log base a (n)
First, let's express 14 as a product of two numbers in powers of 2. It is known that 2^3 = 8, and since 14 is greater than 8, we can write 14 as 8 * 1.75.
Therefore, log base 2 of 14 can be rewritten as:
log base 2 (8 * 1.75)
Now we can use the property mentioned above to simplify it further:
log base 2 (8 * 1.75) = log base 2 (8) + log base 2 (1.75)
Since 2^3 = 8, we can rewrite this as:
log base 2 (2^3) + log base 2 (1.75)
Using another property of logarithms, log base a (a^k) = k, we simplify it to:
3 + log base 2 (1.75)
Now, we need to find the value of log base 2 of 1.75. We don't have this information directly, but we do know that log base 2 of 7 is 2.807.
Let's use a property of logarithms to find log base 2 of 1.75:
log base 2 (1.75) = log base 2 (7/4)
Using the property log base a (m/n) = log base a (m) - log base a (n), we can rewrite it as:
log base 2 (7) - log base 2 (4)
Now, we know that log base 2 of 7 is 2.807, and log base 2 of 4 can be rewritten as log base 2 of (2^2), which is 2. Since we are subtracting it, it becomes -2.
Therefore, log base 2 of 1.75 can be calculated as:
log base 2 (7) - log base 2 (4) = 2.807 - 2 = 0.807
Finally, substituting this value into our initial equation, we get:
3 + log base 2 (1.75) = 3 + 0.807 = 3.807
So, log base 2 of 14 is approximately 3.807.