Write each of the following as a single logarithm:
8) ((5-log x)/4) + 4log y =
To write the given expression as a single logarithm, we can apply the properties of logarithms. Specifically, we can use the rules of logarithms to combine and simplify the expression.
First, let's rewrite the given expression:
((5 - log x)/4) + 4log y
To combine the terms, we can use the property that states: log a + log b = log (ab). This property allows us to group logarithms with the same base together.
Applying the property to the given expression, we get:
(1/4) * (5 - log x) + log y^4
Now, let's simplify further. First, distribute the (1/4) to the terms inside the parentheses:
(1/4) * 5 - (1/4) * log x + log y^4
Simplifying the first term gives:
(5/4) - (1/4) * log x + log y^4
Next, we can apply the property that states: log a^n = n * log a. This allows us to bring the exponent as a coefficient in front of the logarithm.
Applying the property to the last term, we have:
(5/4) - (1/4) * log x + 4 * log y
Finally, we can combine the terms into a single logarithm:
log[((5/4) * y^4) / x]