Calculate Directly
Log (base 4) 2 + log (base 16) 2
since 16=4^2, log_4 = 2log_16
so, converting all to base 4 logs,
log4(2) + 1/2 log4(2)
Since 2 = √4, log4(2) = 1/2, and we have
1/2 + 1/2 (1/2) = 1/2 + 1/4 = 3/4
or, looking at it another way,
2 = √4, so log4(2) = 1/2
2 = ∜16, so log16(2) = 1/4
1/2 + 1/4 = 3/4
To calculate the sum of the logarithms, you can use the logarithmic rule that states:
log (base a) b + log (base a) c = log (base a) (b * c)
Using this rule, we can simplify the expression:
log (base 4) 2 + log (base 16) 2
= log (base 4) 2 * 2 (since 16 can be expressed as 4^2)
= log (base 4) 4
= 1
Therefore, the sum of log (base 4) 2 + log (base 16) 2 is equal to 1.
To calculate the expression, log(base 4) 2 + log(base 16) 2, we can use the logarithmic property known as the "product rule." This rule states that when you add two logarithms with the same base, it is equivalent to taking the logarithm of their product.
In this case, since both logarithms have a base of 2, we can rewrite the expression as:
log(base 4) 2 + log(base 16) 2 = log(base 2) 2 + log(base 2^4) 2
Now, according to another logarithmic property called the "base change rule," we can convert logarithms with different bases by dividing one logarithm by another logarithm with the desired base. Applying the base change rule, we get:
log(base 2) 2 + log(base 2^4) 2 = log(base 2) 2 + log(base 2) 2^4
Since the bases are now the same, we can simplify the expression:
log(base 2) 2 + log(base 2) 2^4 = 1 + 4
Finally, adding the values, we find:
1 + 4 = 5
Therefore, log(base 4) 2 + log(base 16) 2 = 5.