How do you solve $2^x = 5^(x - 2)$ ?

Question

1379 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-07-04T09:48:38

$x=- (2ln(5))/(ln(2)-ln(5)) ~~+3.513$ to 3 decimal places

$color(brown)("Take logs of both sides")$

$" "ln(2^x)=ln(5^(x-2))$

$color(brown)("This is the same as")$

$" "xln(2)=(x-2)ln(5)$

$color(brown)("Divide both sides by "ln(2))$

$" "x=(x-2)(ln(5))/ln(2)$

$color(brown)("Multiply out the bracket")$

$" "x=xln(5)/ln(2)-(2ln(5))/ln(2)$

$color(brown)("Subtract "xln(5)/ln(2)" from both sides")$

$" "x-xln(5)/ln(2)=-(2ln(5))/(ln(2))$

$color(brown)("Factor out the "x)$

$" "x(1-ln(5)/ln(2))=-(2ln(5))/ln(2)$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$color(green)("Note that "(1-ln(5)/ln(2)) = (ln(2)-ln(5))/ln(2))$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$color(brown)("Divide both sides by "(ln(2)-ln(5))/ln(2))$

$" "x=-(2ln(5))/(cancel(ln(2)))xx(cancel(ln(2)))/(ln(2)-ln(5))$

$" "x=- (2ln(5))/(ln(2)-ln(5)) ~~+3.513$ to 3 decimal places

How do you solve 2^x = 5^(x - 2)2^x = 5^(x - 2)?