1. ## Logs

Given that logab(a)=4, calculate logab 3√a /√b. Start with the exponent property of logs, then the quotient rule, and then change of base.

2. Hello KingV15
Originally Posted by KingV15
Given that logab(a)=4, calculate logab 3√a /√b. Start with the exponent property of logs, then the quotient rule, and then change of base.
I'm guessing that the question is:
Given $\displaystyle \log_{ab}(a) = 4$, calculate $\displaystyle \log_{ab}\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right)$
(The alternative is that you have to calculate $\displaystyle \log_{ab}\left(\frac{3\sqrt{a}}{\sqrt{b}}\right)$, which will leave a term in $\displaystyle \log_{ab}(3)$ at the end.)

So, if my assumption is correct:
$\displaystyle \log_{ab}(a) = 4$

$\displaystyle \Rightarrow (ab)^4 = a$

$\displaystyle \Rightarrow a^4b^4 = a$

$\displaystyle \Rightarrow b^4 = a^{-3}$

$\displaystyle \Rightarrow b = a ^{-\frac34}$
And then:
$\displaystyle \log_{ab}\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right)=\tfrac13\log_{ab}(a) - \tfrac12\log_{ab}(b)$
$\displaystyle =\tfrac13\log_{ab}(a) - \tfrac12\log_{ab}(a^{-\frac34})$

$\displaystyle =\tfrac13\log_{ab}(a) - \tfrac12\times(-\tfrac34)\log_{ab}(a)$

$\displaystyle =(\tfrac13+\tfrac38)\log_{ab}(a)$

$\displaystyle =\frac{17}{24}\times 4$

$\displaystyle =\frac{17}{6}$
(The answer to the alternative question is, using a similar method, $\displaystyle \frac72+\log_{ab}(3)$.)