Exponential Equations are equations with a variable exponent.
Example: $9^{5 m-8}=7^{2 m-4}$
Logarithmic Equations are equations which involve a logarithm.
Example: $\log(x-2)-\log(x-5)=\log(x-7)$
Solving Exponential Equations: Type I
Example: Find the exact solution to the equation. Express your answer in terms of base-$e$ logarithms. Then approximate your answer to the nearest tenth. $$9^{6 b+7}=8$$ Example: Find the exact solution to the equation. Express your answer in terms of base-10 logarithms. Then approximate your answer to the nearest tenth. $$9^{5 m-8}=7^{2 m-4}$$
Process: To solve equations of the above form, either $$a^{cx+d}=b^{ex+f},$$ or $$a^{cx+d}=b,$$ 1) Take a logarithm of both sides. Any base will do, but preferably base-10 or base-$e$ since they're built into your calculator.
2) Bring exponents down to earth.
3) Get variables to one side and numbers on the other and solve for $x$.
Solving Exponential Equations: Type II
Example: Find the exact solution to the equation and approximate your answer to the nearest tenth. $$e^{2 q+8}=9$$ Example: Find the exact solution to the equation, and approximate your answer to the nearest tenth. $$5.02e^{0.32 t}=3.43$$
Process: To solve equations of the above form, $$Ae^{cx+d}=b,$$ 1) Isolate $e^{cx+d}$.
2) Take the natural log of both sides, at which point you'll get $$cx+d=\mbox{a decimal number}.$$
3) Get variables to one side and numbers on the other and solve for $x$.
Solving Logarithmic Equations
Example: Solve the logarithmic equation and round your answer to the nearest hundredth. $$\log(6 \theta +7)=-1$$ Example: Solve the logarithmic equation and round your answer to the nearest hundredth. $$\log(x-2)-\log(x-5)=\log(x-7)$$
Process: To solve logarithmic equations
1) Get the equation into one of either two forms: $$\mbox{A) }\,\,\,\,\log_b(\mbox{expression}_1)=\log_b(\mbox{expression}_2), \,\,\,\,\mbox{ or }$$ $$ \mbox{B) }\,\,\,\,\log_b(\mbox{expression})=\mbox{number}$$ 2) To solve equation $\mbox{A) }$, simply set $$\mbox{expression}_1=\mbox{expression}_2.$$ To solve equation $\mbox{B) }$, solve the equation $$\mbox{expression}=b^{\mbox{number}}$$
Dire Warning
Always check for extraneous solutions.
Depreciation The value of certain goods depreciates exponentially.
The value $V$ of a particular model of automobile after $t$ years of depreciation is given by the formula $$V=38000e^{-0.2 t}+1000.$$ Approximately how many years will it take for the value to depreciate to $\$14000$? Round your answer to the nearest hundredth.
