throw some light on logrithmic functions

Yazdan Ahmad Qadri, 13 years ago

Grade:12

1 Answers

suryakanth AskiitiansExpert-IITB

105 Points

13 years ago

Dear Yazdan,

THE LOGARITHMIC FUNCTION WITH BASE b is the function

y = log_b x.

b is normally a number greater than 1 (although it need only be greater than 0 and not equal to 1). The function is defined for all x > 0. Here is its graph for any base b.

Note the following:

• For any base, the x-intercept is 1. Why?

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

The logarithm of 1 is 0. y = log_b1 = 0.

• The graph passes through the point (b, 1). Why?

The logarithm of the base is 1. log_bb = 1.

•	The graph is below the x-axis -- the logarithm is negative -- for

	0 < x < 1.

	Which numbers are those that have negative logarithms?

Proper fractions.

•	The function is defined only for positive values of x.

	log_b(−4), for example, makes no sense. Since b is always positive, no power of b can produce a negative number.

• The range of the function is all real numbers.

• The negative y-axis is a vertical asymptote (Topic 18).

Example 1. Translation of axes. Here is the graph of the natural logarithm, y = ln x (Topic 20).

And here is the graph of y = ln (x − 2) -- which is its translation 2 units to the right.

The x-intercept has moved from 1 to 3. And the vertical asymptote has moved from 0 to 2.

Problem 1. Sketch the graph of y = ln (x + 3).

This is a translation 3 units to the left. The x-intercept has moved from 1 to −2. And the vertical asymptote has moved from 0 to −3.

Please feel free to ask your queries here. We are all IITians and here to help you in your IIT JEE preparation.

All the best.

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