Final answer:
To solve the expression ln(uv^5)^9 = aln(u) + bln(v), we can use the properties of logarithms and equate the coefficients of ln(u) and ln(v). The values of a and b are a = 9 and b = 45.
Step-by-step explanation:
To solve the given expression, we can use the properties of logarithms. First, we can rewrite the expression using the properties ln(a^b) = b ln(a) and ln(A*B) = ln(A) + ln(B). Using these properties, we have:
ln((uv^5)^9) = aln(u) + bln(v)
9 ln(uv^5) = aln(u) + bln(v)
Next, we can use the property ln(uv^a) = ln(u) + a ln(v) to simplify further:
9(ln(u) + 5ln(v)) = aln(u) + bln(v)
Simplifying this equation, we get:
9ln(u) + 45ln(v) = aln(u) + bln(v)
Now, we can equate the coefficients of ln(u) and ln(v):
9 = a
45 = b
Therefore, the values of a and b are a = 9 and b = 45.