How do you express as a single logarithm & simplify (1/2)log_a *x + 4log_a *y - 3log_a *x?

Jul 31, 2015

$\left(\frac{1}{2}\right) {\log}_{a} \left(x\right) + 4 {\log}_{a} \left(y\right) - 3 {\log}_{a} \left(x\right) = {\log}_{a} \left({x}^{- \frac{5}{2}} {y}^{4}\right)$

Explanation:

To simplify this expression, you need to use the following logarithm properties:

$\log \left(a \cdot b\right) = \log \left(a\right) + \log \left(b\right)$ (1)
$\log \left(\frac{a}{b}\right) = \log \left(a\right) - \log \left(b\right)$ (2)
$\log \left({a}^{b}\right) = b \log \left(a\right)$ (3)

Using the property (3), you have:

$\left(\frac{1}{2}\right) {\log}_{a} \left(x\right) + 4 {\log}_{a} \left(y\right) - 3 {\log}_{a} \left(x\right) = {\log}_{a} \left({x}^{\frac{1}{2}}\right) + {\log}_{a} \left({y}^{4}\right) - {\log}_{a} \left({x}^{3}\right)$

Then, using the properties (1) and (2), you have:

${\log}_{a} \left({x}^{\frac{1}{2}}\right) + {\log}_{a} \left({y}^{4}\right) - {\log}_{a} \left({x}^{3}\right) = {\log}_{a} \left(\frac{{x}^{\frac{1}{2}} {y}^{4}}{x} ^ 3\right)$

Then, you only need to put all the powers of $x$
together:

${\log}_{a} \left(\frac{{x}^{\frac{1}{2}} {y}^{4}}{x} ^ 3\right) = {\log}_{a} \left({x}^{- \frac{5}{2}} {y}^{4}\right)$