# If log_a 36=2.2265 and log_a 4=.8614, how do you evaluate log_a 9?

Jun 11, 2015

Using the property of $\log$ which states that ${\log}_{n} \left(\frac{a}{b}\right) = {\log}_{n} a - {\log}_{n} b$

#### Explanation:

Following the $\log$ property, we can use your pieces of information as follows:

$\log 9 = \log \left(\frac{36}{4}\right)$

Thus,

${\log}_{a} 36 - {\log}_{a} 4 = {\log}_{a} 9$

$2.2265 - 0.8614 = {\log}_{a} 9$

${\log}_{a} 9 = 1.3651$

Following $\log$ definition, we have that ${\log}_{a} b = c \iff {a}^{c} = b$

Thus,

${a}^{1.3651} = 9$

To solve this, isolating $a$ we can follow one property of exponentials, which states ${\left({a}^{n}\right)}^{m} = {a}^{n \cdot m}$ by elevating both sides to $\left(\frac{1}{1.3651}\right)$:

${\left({a}^{\cancel{1.3651}}\right)}^{\frac{1}{\cancel{1.3651}}} = {9}^{\frac{1}{1.3651}}$

$a = {9}^{\frac{1}{1.3651}}$

$a \cong 0.7325$