# How do you solve 1/2=10^(1.5x)?

Mar 13, 2016

x = -0.200686663

#### Explanation:

$\frac{1}{2} = {10}^{1.5 \cdot x}$
Take Log on both sides
$\log \left(\frac{1}{2}\right) = 1.5 x \cdot \log 10$
(Hope you are aware of the relationship
$\log {a}^{x} = x \cdot \log a$)
Now $\log \left(\frac{1}{2}\right) = \log {2}^{-} 1 = - 1 \cdot \log 2$
log 2 = 0.3010300 (this you can get from any log tables or calculator)
So, $- 0.3010300 = 1.5 x$
or $x = - \frac{0.3010300}{1.5} = - 0.200686663$
Hope it is clear

Mar 13, 2016

$\therefore x = - \frac{2}{3} {\log}_{10} 2$

#### Explanation:

tking ${\log}_{10}$ on both sides

${\log}_{10} \left(\frac{1}{2}\right) = {\log}_{10} {10}^{1.5 x}$

$\implies {\log}_{10} 1 - {\log}_{10} 2 = {\log}_{10} {10}^{1.5 x}$

$\implies 0 - {\log}_{10} 2 = x \cdot 1.5 \times {\log}_{10} 10$

$\implies 0 - {\log}_{10} 2 = \frac{3 x}{2}$
$\therefore x = - \frac{2}{3} {\log}_{10} 2$