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Logarithms – Solving exponential inequalities

πŸ“˜ Notes

Logarithms β€” Solving Exponential Inequalities

Exponential inequalities are inequalities in which the variable appears in the power. They can be solved by using the same base or by taking logarithms.

1. Main idea

In an exponential inequality, the unknown is in the exponent.

\[ 2^x>8, \qquad 3^{2x-1}\le 27, \qquad 5^x<7 \]

There are two main methods:

  • rewrite both sides using the same base
  • take logarithms of both sides
$$ a^m > a^n \quad \Rightarrow \quad m > n \quad \text{when } a > 1 $$
$$ a^m > a^n \quad \Rightarrow \quad m < n \quad \text{when } 0 < a < 1 $$

If the base is between 0 and 1, the direction reverses.

3. Method 1: Write both sides with the same base

Example: Solve

\[ 2^{x+1}>16 \]

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:

\[ x+1>4 \quad \Rightarrow \quad x>3 \]

4. Example with a base between 0 and 1

Example: Solve

\[ \left(\frac12\right)^x<\left(\frac12\right)^3 \]

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

\[ x>3 \]

5. Method 2: Use logarithms

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

\[ 3^x>7 \]

Take logarithms:

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

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

\[ x>\frac{\log 7}{\log 3} \]

6. Worked example

Example: Solve

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

Take logarithms:

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

Divide by \(\log 5\):

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

\[ x\le \frac{1}{2}\left(\frac{\log 12}{\log 5}+1\right) \]

7. Example with different bases

Example: Solve

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

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:

\[ x>\frac{-\log 5}{\log 2-\log 5} \]

8. Summary of method

  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 β†’ sign stays the same
    • divide by a negative number β†’ sign reverses

9. Exam tips

  • Always check whether the base is greater than 1 or between 0 and 1.
  • Use logarithms when the bases are different and cannot be rewritten neatly.
  • Be very careful when dividing an inequality by a negative number.
  • Keep answers exact in logarithmic form unless a decimal is requested.
  • Write the final answer as an inequality, not just a number.