# Simplify 2^(3+5log_2x)?

Feb 8, 2018

${2}^{3 + 5 {\log}_{2} x} = 8 {x}^{5}$

#### Explanation:

Let ${a}^{{\log}_{a} x} = u$. Then taking logarithm to the base $a$ on both sides we get

${\log}_{a} x \times {\log}_{a} a = {\log}_{a} u$

or ${\log}_{a} u = {\log}_{a} x$ and therefore $u = x$ i.e.

${a}^{{\log}_{a} x} = x$

Using this in ${2}^{3 + 5 {\log}_{2} x}$

= ${2}^{3} \times {2}^{5 {\log}_{2} x}$

= ${2}^{3} \times {\left({2}^{{\log}_{2} x}\right)}^{5}$ - as ${a}^{m n} = {\left({a}^{m}\right)}^{n}$ or ${\left({a}^{n}\right)}^{m}$

= $8 {x}^{5}$