How Can I Understand Logarithms and Their Graphs Better?

[baltimores_finest]

Logarithms...i need help.

A logarithm of a number is the exponent of the power to which a fixed number. called the base, must be raised to produce the given number.

I absolutely do not understand what these things mean. In my textbook it shows a graph of x=10y.
Then it follows with 6 questions which are as follows:
1. For what values of x do the corresponding logarithms change most rapidly?
2. How does the rate of change of y compare with that of x for values of x between 1 and 10?
3. For what values of x are the values of y negative?
4. What is the approximate value of y when x=8? 15? 28?
5. What is the number whose logarithm is .2? .4? 1.2? 1.4?
6. Show that in the graph log 10 is approximately equal to log 5+log 2; that log 5 is approximately equal to log 25-log 5. that log 27 is approximately equal to 3 log 3.

PLEASE ANSWER BUT EXPLAIN HOW THESE ARE DONE PLEASE...

Thanks.
Amber

 

[Janus]

A logarithm of a number is the exponent of the power to which a fixed number. called the base, must be raised to produce the given number.

I absolutely do not understand what these things mean.

I won't answer your questions for you, but I will try to explain logarithms so that you will understand them well enough to do them yourself.

If you take the equation

10^{2}=100

Ten is the base, and 2 is the exponent.

A logarithm is basically solving for x in the following:

10^{x}=100

here x = 2

for

10^{x}= 1000

x=3

Another way of writing this would be

log_{10}1000 = 3

Which would read "The log of 1000, base 10, is 3"

The general form of this equation is

log_{base}(number) = exponent

The exponent (or log of the number) does not have to be a whole number.

Thus, the log of 5, base 10 would be 0.69897 or

log_{10}5 = 0.69897

or of 15:

log_{10}15 = 1.1761

Hope this helps

 

[jcsd]

As your only worried about base 10 logarithms:

10^{log(x)} = x

The above equation all you really need to know for now.

I assume that the graph in the book isn't x = 10y, but x = 10^y, so just by looking at the above equation you should be able to see that y = log(x).

 

[arildno]

A logarithm of a number is the exponent of the power to which a fixed number. called the base, must be raised to produce the given number.

This phrase seems meaningless to me as well!.
I would have said:
A logarithm (with respect to a number B) of a number A is the power to which B must be raised in order to produce A.
The power to which we raise a number is often called the exponent;
the number to be raised is called the base.
If the base is B, the exponent that produces A is called the B-logarithm to A.

Read the other replies carefully; these detail the procedure needed to solve the problems.