Given that log _2 3=m and log _2 5=n,express log_2 30 in terms of m and n
We can use the properties of logarithms to express log_2 30 in terms of m and n.
First, we can write log_2 30 as log_2 (2 * 3 * 5), since 2 * 3 * 5 = 30.
Using the product rule of logarithms, we can split this up into two separate logarithms:
log_2 (2) + log_2 (3) + log_2 (5).
Now, we know that log_2 (2) = 1, since any number raised to the power of 1 is equal to itself.
Using the given information, we can express log_2 (3) as m and log_2 (5) as n.
Therefore, log_2 30 = 1 + m + n.
So, log_2 30 can be expressed in terms of m and n as 1 + m + n.
To express log_2 30 in terms of m and n, we can use the properties of logarithms.
We know that log_2 30 = log_2 (2^m * 5^n) since 30 can be expressed as a product of powers of 2 and 5.
Using the product rule of logarithms, we can break down the expression further:
log_2 30 = log_2 (2^m) + log_2 (5^n)
Now, applying the power rule of logarithms, we can simplify the expression:
log_2 30 = m * log_2 2 + n * log_2 5
Since log_2 2 equals 1, and based on the given information that log_2 3 = m and log_2 5 = n, we can substitute the values:
log_2 30 = m * 1 + n * log_2 5
Therefore, log_2 30 can be expressed as m + n * log_2 5.