What is the answer for this? Write the expression in terms of $\log x$ and $\log y$ $$\log\left(\dfrac{x^3}{10y}\right)$$
This is what I got out of the equation so far. the alternate form assuming $x$ and $y$ are positive $$3\log(x)-\log(10y)$$ or maybe $$\log\left(\dfrac{x^3}{y}\right)-\log(5)=\log(2)$$?
I would appreciate a solution or an input thanks so much.
I think you are close to the expected answer with your first answer. That is, $$\log\left(\frac{x^3}{10y} \right) = \log(x^3) - \log(10y) = 3 \log(x) - \log(10) - \log(y).$$
\begin{align*} \log (\frac{x^3}{10y}) & = \log (x^3) - \log (10y) \\ & = 3 \log x - (\log 10 +\log y) \\ & = 3 \log x - \log 10 - \log y \end{align*} using $$ \log(ab) = \log a + \log b ,$$ $$ \log(\frac{a}{b}) = \log a - \log b $$ and $$ \log(a^n) = n \log a $$
As you have your answer I will simply give some help for future.
There are 3 basic(1,2 and 3) formula regarding logarithm:
1.$\ log_am+\ log_an=\ log_a(mn)$
2.$\ log_am-\ log_an=\ log_a(\dfrac {m}{n})$
3.$\ log_am^n=\ n\times\ log_a(m)$
4.$\ log_aa=1$
5.$\ log_ab\times \ log_na=\log_nb$
6.$\dfrac {\ log_am}{\ log_an}=\ log_nm$
and the basic which useful for solving log equation
$\log_{a}{b}=n\Longleftrightarrow a^n=b$
These can solve many problems regarding logarithm.
