# How do you solve log_9(x^7)=15 ?

Dec 16, 2015

$x = {9}^{\frac{15}{7}}$

#### Explanation:

From the definition of a logarithm, we have

${a}^{{\log}_{a} \left(x\right)} = x$

Applying that here, we get

${\log}_{9} \left({x}^{7}\right) = 15$

$\implies {9}^{{\log}_{9} \left({x}^{7}\right)} = {9}^{15}$

$\implies {x}^{7} = {9}^{15}$

$\implies x = {\left({9}^{15}\right)}^{\frac{1}{7}} = {9}^{\frac{15}{7}}$

Dec 17, 2015

$x = {9}^{\frac{15}{7}}$

#### Explanation:

Use the logarithm rule: ${\log}_{a} \left({b}^{c}\right) = c \cdot {\log}_{a} \left(b\right)$

Thus, the equation can be rewritten as

$7 {\log}_{9} \left(x\right) = 15$

Divide both sides by $7$

${\log}_{9} \left(x\right) = \frac{15}{7}$

Undo the logarithm

9^(log_9(x)=9^(15/7)

$x = {9}^{\frac{15}{7}}$