Find the Real Root of 4^x + 6^x = 9^x

Given a equation $tex_920788608923e1a29dcc453bcaa190e7 Find the Real Root of 4^x + 6^x = 9^x math$ , let’s find the real value of x.

First, divide both sides by $tex_3713d39bf3977004053c53c30e5c84e4 Find the Real Root of 4^x + 6^x = 9^x math$ which becomes:

$tex_85d895cc0e7faade1d6b803a6d2d94f5 Find the Real Root of 4^x + 6^x = 9^x math$
as $tex_1ac70cc9e9bf75d0f3cdc56625c5d08d Find the Real Root of 4^x + 6^x = 9^x math$ , and $tex_b3d6ca55e09f520d99de7976e24efaba Find the Real Root of 4^x + 6^x = 9^x math$ , it becomes:

$tex_6acc2c885bd3873669b5aeada000a14a Find the Real Root of 4^x + 6^x = 9^x math$

Let’s replace $tex_ab7c8bd1f7fbb973aa58d541a7f65f78 Find the Real Root of 4^x + 6^x = 9^x math$ with $tex_08d76dd2192e1355ed65b2f013af4415 Find the Real Root of 4^x + 6^x = 9^x math$ and further we have:

$tex_d576ee8ed6e66bc97dfbdb0e3c923598 Find the Real Root of 4^x + 6^x = 9^x math$ that is $tex_dcdf980b45bf9707b52e3d2ac74fb84f Find the Real Root of 4^x + 6^x = 9^x math$

This equation has a golden root (the negative root should be discarded):

$tex_84f336b54e94a3f6a1141784af0afa4f Find the Real Root of 4^x + 6^x = 9^x math$

Therefore, $tex_6a0d8158137223f60381ab35de616748 Find the Real Root of 4^x + 6^x = 9^x math$
and $tex_57c96216012ab243d3db15aa5ddff79c Find the Real Root of 4^x + 6^x = 9^x math$
and $tex_bb249fb49321758ad7d8db33e6550647 Find the Real Root of 4^x + 6^x = 9^x math$
and we can compute the value x that is approximately 1.187.