03Eexpeqnsbase(e)

toc = Exponential equations (base e) =


 * Euler's number, **//e,//** is an irrational number.

math . \quad \centerdot \;\; e \approx 2.71828 math
 * 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 opposite function to //e// x is the **natural logarithmic function**.
 * Notation for the natural log is **//log// e (x)** or **//ln//(x)**.

math \text{If}\quad\;e^x=a\quad\text{then}\quad \log_ea=x \quad \text{or} \quad \ln(a)=x math

On your calculator, you can enter e x from the virtual keyboard. Go to the **MTH** tab or the **2D** tab {you will also see LOG (base 10) and LN (base e) in the same place}

**Example 1**
... Solve the following for x, showing
 * (i) an exact answer and
 * (ii) an answer correct to 3 decimal places.

math \\ . \qquad (a) \quad 3e^{2x}=9 \\. \\ . \qquad (b) \quad e^x-5e^{-x}=4 math


 * __Solution:__**

math \\ . \qquad 3e^{2x}=9 \\. \\ . \qquad e^{2x}=3 \;\; \iff \;\; \log_e3=2x \\. \\ . \qquad 2x = \log_e 3 \\. \\ . \qquad x=\frac{1}{2} \log_e3 \qquad \textit{exact answer} math . math . \qquad x = 0.549 \qquad \textit{correct to 3 decimal places} math
 * (a) **

math . \qquad e^x-5e^{-x}=4 math . multiply both sides by e x math \\ . \qquad e^{2x}-5=4e^x \\. \\ . \qquad (e^x)^2-4e^x-5=0 math . Let u = e x math \\ . \qquad u^2-4u-5=0 \\. \\ . \qquad (u-5)(u+1) = 0 \\. \\ . \qquad u=5 \;\; \textit{ or } \;\; u=-1 math . but u = e x math . \qquad e^x=5 \;\; \textit{ or } \;\; e^x=-1 math
 * (b) ** [[image:03Eeg1.gif width="305" height="456" align="right"]]

e x = –1 is impossible since e x > 0 for all values of x

math \\ . \qquad e^x=5 \;\; \iff \;\; x= \log_e5 \\. \\ . \qquad x= \log_e5 \qquad \textit{exact solution } math . math . \qquad x=1.609 \qquad \textit{correct to 3 decimal places} math

media type="custom" key="8199148" Some equations can only be solved on the calculator

** Example 2 **
Solve using a calculator, e 2x = 3x + 2media type="custom" key="8199234" align="right"

{click on linkes to see steps} Using the media type="custom" key="8199164" width="80" height="80" __may__ only give one solution: x = –0.5573 But it also gives a warning that ** "More solutions may exist" **
 * __Solution__**

To overcome this, we must media type="custom" key="8199190" width="80" height="80" the two equations: y = e 2x and y = 3x + 2

{**ZOOM** as appropriate, in this case: "**Quick e^x"** works well}

The graph clearly shows two intercepts. The __x-coodinates__ of those intercepts will be the solutions to our equation.

To find the intercepts, Go to the ANALYSIS menu, GSOLVE submenu and select media type="custom" key="8199218" width="80" height="80".

This confirms our first solution: x = –0.5573

Use the __side arrows__ to swap to the media type="custom" key="8199220" width="80" height="80".

This gives the other solution: x = 0.7086

Hence the solutions are {x = –0.5573, x = 0.7086}

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