Hi, an attachment is the best way that I thought I could present this problem. Please take a look.
Hello,Originally Posted by coopsterdude
an attachment is the best way to give you first fundamental informations. Please take a look: In the left column of the table there are the rules how to calculate power values; in the right column there are the corresponding rules how to calculate logarithm values.
(I am very much in a hurry. I hope that I'll find some time this afternoon to give you more detailed informations. But maybe the attachment helps a little bit further with your studies.)
Greetings
EB
Hello,Originally Posted by coopsterdude
here I am again:
I presume that you know to calculate powers of a number
I presume too, that you know the most important rules in calculating powers:
$\displaystyle a^n \cdot a^m=a^{n + m}$ and
$\displaystyle a^n / a^m=a^{n - m}$
That means that you reduce the multiplication of numbers to the addition of the exponents. The greek word oft the Latin word exponent is logarithm. So logarithm is exactly the same as an exponent.
And that means too, that you reduce the division of numbers to the subtraction of the exponents. Look at the given examples of your book!
To use logarithms you have to name the base explicitly to which this logarithm belongs:
$\displaystyle 2^3=8$ means that 3 is the logarithm, which belong to the base 2 so that you obtain the result 8. That's a pretty long sentence, so mathematicians abbreviated it to:
$\displaystyle \log_{2} 8=3$. If you forget to name the base you could maybe write $\displaystyle 5^3=125$ and you will not get 8 but another number, depending on which base you chose although the exponent is still 3.
To use logarithms you have to know the exponents of every positive real number belonging to one base. Fortunately your calculator know the exponents of all (positive) numbers to the base $\displaystyle e \approx 2.718281828...$, which were called natural logarithms and you'll find them under the button ln. And your calculator "knows" too the exponents of all (positive) numbers to the base 10, which were called decimal (or Brigg's) logarithms and you'll find them under the button log. That means $\displaystyle \log(2)=\log_{10}(2) \approx 0.30103$.
With these logarithms you can calculate the logarithm of a number to any base you chose:
Let the number be x and the base you have chosen is b. Then you get:
$\displaystyle b^{\log_{b}{x}}=x$.
Now you calculate the logarithms of b and x to the base 10 (remember: your calculator knows these logarithms):
$\displaystyle \left( 10^{\log_{10}{b}}right)^{\log_{b}{x}}=10^{\log_{10 }{x}}$
Now you've an equation which consists of two powers to the base 10. This equation is true if the exponents are equal. So you get:
$\displaystyle \log_{10}{b}} \cdot \log_{b}{x}}=\log_{10}{x}$.
Divide both sides of this equation by $\displaystyle \log_{10}{b}}$ and you'll get a formula for calculating logarithms to any base:
$\displaystyle \log_{b}{x}}=\frac{\log_{10}{x}}{\log_{10}{b}}}$
To demonstrate how to use this formula: Calculate the $\displaystyle \log_{\sqrt{3}}{81}=\frac{\log(81)}{\log(\sqrt{3}) }=8\ \mbox{because}\ \left(\sqrt{3} \right)^8=81$
Compare the given examples with the tables of rules I've send in my first post and use the hints of this post I'm pretty certain that you easily detect how to use logarithm.
Greetings
EB