Solving Inequalities with Exponents: Maximizing x

[ubergewehr273]

Homework Statement

$81^5>32^x$
Find the maximum value of $x$ in order to satisfy the inequality.

Homework Equations

Inequalities, indices

The Attempt at a Solution

Try to make the bases on both sides of the inequality same.

 

[ShayanJ]

You can't make the bases equal because $81=3^4$ and $32=2^5$. You should solve $81^5=32^x$ for x. That'll be the maximum value! Just get the logarithm(in any base) of both sides. That'll get out x in a way that you can isolate it.

 

[Mark44]

You can't make the bases equal because $81=3^4$ and $32=2^5$.

Actually, you can make the bases equal.
81⁵ = (3⁴)⁵ = 3²⁰, and $32 = 3^{log_3(32)}$

You should solve $81^5=32^x$ for x. That'll be the maximum value! Just get the logarithm(in any base) of both sides. That'll get out x in a way that you can isolate it.

To the OP:
In future posts, you need to make more of an effort than this.

Try to make the bases on both sides of the inequality same.

 

[Joffan]

You can reduce it somewhat:

$81^5 > 32^x = 2^{5x}$
∴ ...