Processing Math: Done
To print higher-resolution math symbols, click the
Hi-Res Fonts for Printing button on the jsMath control panel.

No jsMath TeX fonts found -- using image fonts instead.
These may be slow and might not print well.
Use the jsMath control panel to get additional information.
jsMath Control PanelHide this Message


jsMath

Solution 3.4:1a

From Förberedande kurs i matematik 1

(Difference between revisions)
Jump to: navigation, search
(Ny sida: {{NAVCONTENT_START}} <center> Bild:3_4_1a-1(2).gif </center> {{NAVCONTENT_STOP}} {{NAVCONTENT_START}} <center> Bild:3_4_1a-2(2).gif </center> {{NAVCONTENT_STOP}})
Current revision (08:45, 2 October 2008) (edit) (undo)
m
 
(3 intermediate revisions not shown.)
Line 1: Line 1:
-
{{NAVCONTENT_START}}
+
We solve an equation of this type by taking the natural logarithm of both sides
-
<center> [[Bild:3_4_1a-1(2).gif]] </center>
+
 
-
{{NAVCONTENT_STOP}}
+
{{Displayed math||<math>\ln e^x = \ln 13\,\textrm{.}</math>}}
-
{{NAVCONTENT_START}}
+
 
-
<center> [[Bild:3_4_1a-2(2).gif]] </center>
+
Then, using the log law <math>\ln a^{b} = b\cdot \ln a</math>, the unknown <math>x</math> can then be moved down as a factor on the left-hand side
-
{{NAVCONTENT_STOP}}
+
 
 +
{{Displayed math||<math>x\cdot \ln e = \ln 13</math>}}
 +
 
 +
and then it is simple to solve for <math>x</math>,
 +
 
 +
{{Displayed math||<math>x = \frac{\ln 13}{\ln e} = \frac{\ln 13}{1} = \ln 13\,\textrm{.}</math>}}
 +
 
 +
 
 +
Note: One thing that we haven't mentioned in this solution is that before we carry out the first step and take the logarithm of both sides of the equation, it is necessary to be certain that the left- and right-hand sides are positive (it is not possible to take the logarithm of a negative number). Because the equation is
 +
 
 +
{{Displayed math||<math>e^x=13</math>}}
 +
 
 +
we see directly that the right-hand side is positive. The left-hand side is also positive because “a positive number <math>e\approx 2\textrm{.}718</math> raised to a number” always gives a positive number.

Current revision

We solve an equation of this type by taking the natural logarithm of both sides

lnex=ln13.

Then, using the log law lnab=blna, the unknown x can then be moved down as a factor on the left-hand side

xlne=ln13

and then it is simple to solve for x,

x=lneln13=1ln13=ln13.


Note: One thing that we haven't mentioned in this solution is that before we carry out the first step and take the logarithm of both sides of the equation, it is necessary to be certain that the left- and right-hand sides are positive (it is not possible to take the logarithm of a negative number). Because the equation is

ex=13

we see directly that the right-hand side is positive. The left-hand side is also positive because “a positive number e2.718 raised to a number” always gives a positive number.