Solving Exponential Equations- Basic - Maple Programming Help

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Solving Exponential Equations- Basic

Main Concept

Basic techniques for solving equations with exponentials

The two most useful techniques for solving equations involving exponentials are:

 1 Look for a common base; and
 2 Look for a common exponent.

The first enables you to manipulate terms in the equation using the rules ${a}^{x}{a}^{y}={a}^{x+y}$,  $\frac{{a}^{x}}{{a}^{y}}={a}^{x-y}$ and ${\left({a}^{x}\right)}^{y}={a}^{x\cdot y}$. The second lets you take advantage of the rule ${a}^{x}{b}^{x}={\left(a\cdot b\right)}^{x}$. It is often the case that the two rules are used in tandem; for example, if $c=a\cdot b$ then $\frac{{a}^{x}}{{c}^{x}}=\frac{{a}^{x}}{{\left(a\cdot b\right)}^{x}}=\frac{{a}^{x}}{{a}^{x}{b}^{x}}={b}^{-x}$.

Example: Find $x$ such that .

Solution: Note that 8 and 4 can both be expressed as powers of 2 ($8={2}^{3}$ and $4={2}^{2})$, so , so $x=-1$.

 Using logarithms to solve equations with exponentials See the lessons on logarithms elsewhere in this collection for information regarding the properties of logarithms.   An equation which can be recast into the form ${a}^{x}=b$ can be solved for $x$ in terms of logarithms: $x={\mathrm{log}}_{a}\left(b\right)$.     Example: Find $x$ such that . Solution: Since $9={3}^{2}$ we have . Multiplying through by , we get       Taking the base-3 logarithm of both sides, we have , so, finally, $x=\frac{1}{5}{\mathrm{log}}_{3}\left(17\right)$.

The graph shows two exponential functions of the form $c\mathit{\cdot }{a}^{r\mathit{\cdot }x}$. Use the sliders to change the parameters $c$, $a$, and $r$ for each function. The point of intersection of the graphs of the two functions, which is the point at which the two functions are equal, is shown as a magenta dot. Experiment with the sliders to see how the shapes of the graphs, and hence their points of intersection, are affected by each of the parameters. In particular, observe what happens when the value of one of the $a$ parameters changes from less than 1 to greater than 1, or when one of the $r$ parameters changes from positive to negative, and vice versa.



   



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