Solving Exponential Equations (DP IB Analysis & Approaches (AA): SL): Revision Note

Solving exponential equations

What are exponential equations?

• An exponential equation is an equation where the power is the unknown

  • $5^x=8$

  • $7=2^{x+3}$

How do I solve exponential equations?

• Some exponential equations can be solved by inspection (seeing the answer)

  • e.g. $3^x=9$ is solved by $x=2$

• Others can be solved by changing the base

  • e.g. $3^{x+1}=\frac{1}{9^x}$ is solved by writing $9$ as $\left(3^2\right)$:

$\begin{array}{rcl}{3}^{x+1}& =& \frac{1}{{\left(3^2\right)}^x}\\ 3^{x+1}& =& \frac{1}{3^{2x}}\\ 3^{x+1}& =& 3^{-2x}\\ x+1& =& -2x\\ 3x& =& -1\\ x& =& -\frac{1}{3}\end{array}$

• But some exponential equations have no obvious answers, in which case use logarithms

  • e.g. $2^x=5$ so $x={\log }_{2}5=2.321928...$

    • Recall if $a^x=b$ then $x={\log }_a\left(b\right)$

How do I solve quadratic exponential equations?

• A quadratic exponential equation is ${\left(a^x\right)}^2+b\left(a^x\right)+c=0$

  • This is a hidden quadratic in $a^x$

    • You can solve it with the substitution $u=a^x$

    • This gives two separate exponential equations to solve

• Beware: the ${\left(a^x\right)}^2$ term may be disguised as $a^{2x}$

  • Use the index law $a^{2x}={\left(a^x\right)}^2$ to correct this:

    • ${\mathrm{e}}^{2x}={\left({\mathrm{e}}^x\right)}^2$

    • $3^{2x}={\left(3^x\right)}^2$

    • ${16}^x=4^{2x}={\left(4^x\right)}^2$ (a change of base is also needed here)