Question 367073
{{{y=1000(2^(-0.1x))}}}
It help to rewrite this with a positive exponent. In general, {{{a^(-p) = 1/a^p}}}. so your expression can be rewritten as:
{{{y=1000(1/2^(0.1x))}}}<br>
if x = 10...
{{{y=1000(1/2^(0.1(10)))}}}
which simplifies as follows:
{{{y=1000(1/2^1)}}}
{{{y=1000(1/2)}}}
{{{y = 500}}}<br>
if y = 250...
{{{250=1000(1/2^(0.1x))}}}
Divide both sides by 1000:
{{{250/1000=(1000(1/2^(0.1x)))/1000}}}
which simplifies to:
{{{1/4=1/2^(0.1x)}}}
This is a proportion so we can cross multiply:
{{{1*2^(0.1x) = 4*1}}}
or
{{{2^(0.1x) = 4}}}
The shortest way to solving for x here is to recognize that both sides of the equation are powers of two:
{{{2^(0.1x) = 2^2}}}
(Note: If both sides had not both been powers of the same number, then we would have to use logarithms to solve for x.)
The only way for powers of 2 to be equal is if their exponents are equal. So:
{{{0.1x = 2}}}
And finally we can multiply both sides by 10:
{{{10(0.1x) = 10(2)}}}
which simplifies to:
{{{x = 20}}}