# How do you solve log_10y = (1/4)log_10(16) + (1/2)log_10(25)?

Dec 21, 2015

$y = 10$

#### Explanation:

${\log}_{10} \left(y\right) = \left(\frac{1}{4}\right) {\log}_{10} \left(16\right) + \left(\frac{1}{2}\right) {\log}_{10} \left(25\right)$

Rules which can be used here.

1. $n \cdot \log \left(a\right) = \log \left({a}^{n}\right)$
2. $\log \left(a\right) + \log \left(b\right) = \log \left(a b\right)$
3. If $\log \left(a\right) = \log \left(b\right)$ then $a = b$

${\log}_{10} \left(y\right) = {\log}_{10} {\left(16\right)}^{\frac{1}{4}} + {\log}_{10} {\left(25\right)}^{\frac{1}{2}}$ By rule 1.
${\log}_{10} \left(y\right) = {\log}_{10} \left(2\right) + {\log}_{10} \left(5\right)$ since ${a}^{\frac{1}{n}} = \sqrt[n]{a}$
${\log}_{10} \left(y\right) = {\log}_{10} \left(2 \cdot 5\right)$ By rule 2.
${\log}_{10} \left(y\right) = {\log}_{10} \left(10\right)$

$y = 10$ By Rule 3.

$y = 10$ is the final answer.