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 3.3:6a

From Förberedande kurs i matematik 1

(Difference between revisions)
Jump to: navigation, search
m (Lösning 3.3:6a moved to Solution 3.3:6a: Robot: moved page)
Line 1: Line 1:
-
{{NAVCONTENT_START}}
+
The calculator does not have button for
-
<center> [[Image:3_3_6a-1(2).gif]] </center>
+
<math>\log _{3}</math>, but it does have one for the natural logarithm ln, so we need to rewrite
-
{{NAVCONTENT_STOP}}
+
<math>\log _{3}4</math>
-
{{NAVCONTENT_START}}
+
in terms of ln.
-
<center> [[Image:3_3_6a-2(2).gif]] </center>
+
 
-
{{NAVCONTENT_STOP}}
+
If we go back to the definition of the logarithm, we see that
-
[[Image:3_3_6_a.gif|center]]
+
<math>\log _{3}4</math>
 +
is that number which satisfies
 +
 
 +
 
 +
<math>3^{\log _{3}4}=4</math>
 +
 
 +
 
 +
Now, take the natural logarithm of both sides,
 +
 
 +
 
 +
<math>\ln 3^{\log _{3}4}=\ln 4</math>
 +
 
 +
 
 +
Using the logarithm law,
 +
<math>\lg a^{b}=b\lg a</math>, the left-hand side can be written as
 +
<math>\log _{3}4\centerdot \ln 3</math>
 +
and the relation is
 +
 
 +
 
 +
<math>\log _{3}4\centerdot \ln 3=\ln 4</math>
 +
 
 +
 
 +
Thus, after dividing by
 +
<math>\text{ln 3}</math>, we have
 +
 
 +
 
 +
<math>\log _{3}4=\frac{\ln 4}{\ln 3}=\frac{1.386294...}{1.098612...}=1.2618595</math>
 +
 
 +
 
 +
which gives 1.262 as the rounded-off answer.
 +
 
 +
NOTE: on a calculator, the answer is obtained by pressing the buttons
 +
 
 +
 
 +
<math>\begin{align}
 +
& \left[ 4 \right]\quad \left[ \text{LN} \right]\quad \left[ \div \right]\quad \left[ 3 \right]\quad \left[ \text{LN} \right]\quad \left[ = \right] \\
 +
& \quad \left[ 4 \right]\quad \left[ \text{LN} \right]\quad \left[ \div \right]\quad \left[ 3 \right]\quad \left[ \text{LN} \right]\quad \left[ = \right] \\
 +
\end{align}</math>

Revision as of 09:08, 26 September 2008

The calculator does not have button for log3, but it does have one for the natural logarithm ln, so we need to rewrite log34 in terms of ln.

If we go back to the definition of the logarithm, we see that log34 is that number which satisfies


3log34=4


Now, take the natural logarithm of both sides,


ln3log34=ln4


Using the logarithm law, lgab=blga, the left-hand side can be written as log34ln3 and the relation is


log34ln3=ln4


Thus, after dividing by ln 3, we have


log34=ln3ln4=10986121386294=12618595


which gives 1.262 as the rounded-off answer.

NOTE: on a calculator, the answer is obtained by pressing the buttons


4LN3LN=4LN3LN= 

Retrieved from "http://wiki.math.se/wikis/2008/bridgecourse1-ImperialCollege/index.php/Solution_3.3:6a"