Example: Solve

Write \(16\) as a power of 2:

\[ 2^{x+1}>2^4 \]

Since the base \(2\) is greater than 1, the inequality stays the same:

Example: Solve

Since the base \(\frac12\) is between 0 and 1, the inequality reverses:

If the inequality cannot easily be written using the same base, take logarithms of both sides.

Since logarithms with base greater than 1 are increasing functions, the inequality direction stays the same.

Example: Solve

Take logarithms:

\[ \log(3^x)>\log 7 \] \[ x\log 3>\log 7 \]

Since \(\log 3>0\), divide by \(\log 3\) without reversing the sign:

Example: Solve

Take logarithms:

\[ \log(5^{2x-1})\le \log 12 \] \[ (2x-1)\log 5\le \log 12 \]

Divide by \(\log 5\):

\[ 2x-1\le \frac{\log 12}{\log 5} \] \[ 2x\le \frac{\log 12}{\log 5}+1 \]

Example: Solve

Take logarithms:

\[ \log(2^x)<\log(5^{x-1}) \] \[ x\log 2<(x-1)\log 5 \]

Expand and collect \(x\)-terms:

\[ x\log 2

Since \(\log 2-\log 5<0\), dividing by it reverses the inequality:

1. Check whether both sides can be written in the same base.
2. If yes, compare exponents carefully.
3. If not, take logarithms of both sides.
4. When dividing an inequality, remember:
   • divide by a positive number → sign stays the same
   • divide by a negative number → sign reverses