how to write each expression into a single logarithm.
3log[4]x^(2) + (1/2)log[4]�ã(x)
the answer is: (25/4)log[4]x
steps please
I will assume the strange symbol is a square root sign
3log[4]x^(2) + (1/2)log[4]?ã(x)
=log4x^6 + log4x^(1/4)
= log4 ((x^6)(x^(1/4))
= log4 (x^(25/4)
= (25/4)log4 x
To write each expression into a single logarithm, we can use the properties of logarithms. Remembering the property that states log(a) + log(b) = log(ab), we can apply it to the given expressions.
Step 1: Start with the first expression.
3log[4]x^(2)
Since we have a coefficient of 3 in front of the logarithm, we can rewrite it as the product of x^2 raised to the power of 3. Using the property mentioned earlier, we have:
3log[4]x^(2) = log[4](x^(2))^3
= log[4]x^(6)
Step 2: Move to the second expression.
(1/2)log[4]�ã(x)
Similar to the first expression, we have a coefficient of 1/2. We can rewrite it as the logarithm of x raised to the power of 1/2. Applying the property, we get:
(1/2)log[4]�ã(x) = log[4](x)^(1/2)
= (1/2)log[4]x
Step 3: Combine the two expressions.
Now, we can combine the two expressions we obtained in Step 1 and Step 2 by using the property of addition, log(a) + log(b) = log(ab).
log[4]x^(6) + (1/2)log[4]x
Now, we have a common base of 4 and two logarithms to add together.
To add two logarithms with the same base, we can apply another property, which states that log(a) + log(b) = log(ab).
Therefore, we can rewrite the expression as:
log[4](x^(6)(x)^(1/2))
= log[4](x^(6 + 1/2))
= log[4](x^(12/2 + 1/2))
= log[4](x^(25/2))
Finally, we simplify the expression to its simplest form:
log[4](x^(25/2))
= (25/2)log[4]x
Thus, the final answer is (25/2)log[4]x, which can also be written as (25/4)log[4]x.