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

Question

1337 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-29T22:03:39

How you do it is explained below.

Take the natural logarithm of both sides:

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

Use the identity $ln(a^b) = ln(a)b$

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

Distribute ln(5):

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

Subtract ln(5)x from both sides:

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

Multiply both sides by -1:

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

Divide both sides by $(ln(5)-ln(2))$ :

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

$x ~~ 3.5$

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