Simplify the following expression
5lnx - 1/2 lny + 3 lnz
Since
ln(a)+ln(b) = ln(ab)
ln(a)-ln(b) = ln(a/b)
(1/2)ln(a) = ln(sqrt(a), and
3ln(a) = ln(a3)
5 ln(x) = ln(x5)
-(1/2)ln(y) = ln(1/sqrt(y))
3 ln(z) = ln(z3)
So
5lnx - 1/2 lny + 3 lnz
=ln(x5 * 1/sqrt(y) * ... )
Can you now complete the answer and simplify?
Mathmate - Yes, I can thank you!
To simplify the expression 5lnx - 1/2lny + 3lnz, we can use logarithm rules.
1. Start by using the power rule, which states that ln(a^b) = bln(a). Rewrite the expression using this rule:
5ln(x) - ln(y^(1/2)) + 3ln(z)
2. Next, apply the rule for the logarithm of a product, which states that ln(a * b) = ln(a) + ln(b):
ln(x^5) - ln(y^(1/2)) + ln(z^3)
3. Applying the rule for the logarithm of a quotient, which states that ln(a / b) = ln(a) - ln(b), simplify further:
ln((x^5 * z^3) / y^(1/2))
4. Finally, use the rule for the logarithm of a power, which states that ln(a^b) = bln(a):
ln((x^5 * z^3) / √y)
And that's the simplified expression: ln((x^5 * z^3) / √y)