04 51indexeqnse

Exponential equations (base e)

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  • Euler's number, e, is an irrational number. $e \approx 2.71828$

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  • e is named after Leonard Euler a Swiss mathematician (18th Century)
  • a more exact value of e can be calculated in several ways (see here) {not part of course}
  • $e^x$ is a very important function in mathematics and is called the natural exponential function.
  • The index laws apply to e as with any other base.
  • The inverse function to $e^x$ is the natural logarithmic function.
    • Notation for the natural log is $\log_e(x) \; \text{ or } \; \text{ ln}(x)$.
  • $e^x=a \;\; \iff \;\; \log_e(a)=x$

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On the Classpad, use the ex button from the virtual keyboard.04.51icon.gif
* do not use the e from the alphabet or variable list

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Example 1

Solve the following for x, showing an exact answer and an answer correct to 3 decimal places.

a)$3e^{2x}=9$

b)$e^x-5e^{-x}=4$

Solution:

a)$3e^{2x}=9$

… … … $e^{2x}=3 \;\; \iff \;\; \log_e(3)=2x$

… … … $2x = \log_e(3)$

… … … $x=\frac{1}{2} \log_e(3)$ … … exact answer

… … … $x = 0.549$ … … … correct to 3 decimal places
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04.51eg1.gif

b)$e^x-5e^{-x}=4$

… … … multiply both sides by $e^x$

… … … $e^{2x}-5=4e^x$

… … … $(e^x)^2-4e^x-5=0$

… … … Let $u = e^x$

… … … $u^2-4u-5=0$

… … … $(u-5)(u+1) = 0$

… … … $u=5 \;\; \text{ or } \;\; u=-1$

… … … Substitute $u = e^x$

… … … $e^x=5 \;\; \text{ or } \;\; e^x=-1$

… … … Reject $e^x = -1$ since $e^x > 0$ for all values of x

… … … $e^x=5 \;\; \iff \;\; x= \log_e(5)$

… … … $x= \log_e(5)$ … … exact solution

… … … $x=1.609$ … … … correct to 3 decimal places

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Some equations can only be solved on the calculator

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Example 2

04.51eg2a.gif

Solve using a calculator, $e^{2x} = 3x + 2$

Solution

… … Using the solve function may only give one solution: $x = –0.5573$

… … Or it may give two solutions $\{ x = -0.5573, \; x = 0.7086 \}$

… … But it also gives a warning that "More solutions may exist"

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… … To overcome this, we must graph the two equations:

… … $y = e^{2x} \; \text{ and } \; y = 3x + 2$

… … ZOOM as appropriate, in this case: "Quick e^x" works well

04.51eg2b.gif

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… … The graph clearly shows two intercepts,
… … and because of the shape of the graphs, there will not be any more.

… … The x-coordinates of those intercepts will be the solutions to our equation.

… … To find the intercepts, Go to the ANALYSIS menu, GSOLVE submenu and select intersect.

… … This confirms our first solution: $x = -0.5573$

… … Use the side arrows to swap to the other solution.

… … This gives the other solution: $x = 0.7086$

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… … Hence the solutions are $\{ x = -0.5573, \; x = 0.7086 \}$

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