Exponential and Logarithmic Functions
5.7 Exp. and Log. Equations
Solving Exponential Equations
Use matching bases when possible. If not, use logarithms.
Theorem: One-to-one property of exponentials
If $b^u=b^v$ with $b>0$ and $b\ne1$, then
$$u=v.$$
Example: Solve by matching bases
Solve:
$$3^{2x-1}=27.$$
Show Solution
Write $27=3^3$:
$$3^{2x-1}=3^3\Rightarrow 2x-1=3\Rightarrow 2x=4\Rightarrow x=2.$$
Example: Solve using logarithms
Solve:
$$5^{x}=17.$$
Show Solution
Take log of both sides:
$$x\log 5=\log 17\Rightarrow x=\frac{\log 17}{\log 5}. $$
Approximation:
$$x\approx1.7604.$$
MyOpenMath: Practice: exponential equations (placeholder)
Solving Logarithmic Equations
Combine logs first when helpful, and always check domain restrictions.
Example: Solve a logarithmic equation
Solve:
$$\log(x-1)+\log(x-3)=1.$$
Show Solution
Domain requires $x-1>0$ and $x-3>0$, so $x>3$.
Combine logs:
$$\log\big((x-1)(x-3)\big)=1.$$
Convert to exponential form (base 10):
$$(x-1)(x-3)=10.$$
Expand:
$$x^2-4x+3=10\Rightarrow x^2-4x-7=0.$$
Solve:
$$x=2\pm\sqrt{11}. $$
Check domain $x>3$: only
$$x=2+\sqrt{11}$$
is valid.
