In this page, you can find the complete solutions of the fourth exercise of Relations, Functions, and Graphs chapter from Basic Mathematics Grade XI book published and distributed by Sukunda Pustak Bhawan.

In the above-mentioned book, Relations, Functions, and Graphs is the 2nd chapter and has five exercises only. Out of which, this is the solution of the fourth exercise.

Check: Basic Mathematics Grade 11 (Sukunda Publication) Guide:
Grade 11 Basic Mathematics by Sukunda Pustak Vawan Notes and Solutions | Nepal

Disclaimer:

Answers mentioned here are not solved by teachers. These are the solutions written by a student of Grade 11. Answers are all correct. However, the language or process of solving the questions might be informal and in examinations, you might have to add little more language and show more calculations than what has been done here. So, we highly encourage you to view these solutions as guide rather than just copying everything mentioned here. Few questions have been typed while most of them have been updated as pictures.

1. Prove that

a) $\rm \log_a(xy^3/z^2) = \log_a x \log_a y - 2 \log_a z$

Solution:
Given,

$\rm =  \log_a(xy^3/z^2)$

Using the quotient rule of logarithm

$\rm = \log_a(xy^3) - \log_a z^2$

Using the product rule of logarithm

$\rm = \log_a (x) + \log_a(y^3) - \log_a z^2$

Using the power rule of logarithm

$\rm = \log_a (x) + 3 \log_a(y) - 2 \log_a(z)$


b) $\rm \log_a 2x + 3( \log_a x - \log_a y) = \log_a (2x^4/y^3)$

Solution:
Given,

$\rm = \log_a 2x + 3 ( \log_a x - \log_a y)$

$\rm = \log_a 2x + 3 \log_a \left ( x/y \right )$

$\rm = \log_a 2x + \log_a ( x/y)^3$

$\rm = \log_a 2x + \log_a (x^3/y^3)$

$\rm = \log_a ( 2x \times x^3/y^3)$

$\rm = \log_a (2x^4/y^3)$

c) $\rm \log_a x^2 - 2 \log_a \sqrt{x} = \log_a x$

Solution:
Given,

$\rm = \log_a x^2 - 2 \log_a \sqrt{x}$

$\rm = \log_a x^2 - 2 \log_a (x)^{\frac{1}{2}}$

Using the power rule of logarithm

$\rm = 2 \log_a x  - 2 \times \frac{1}{2} \log_a x$

$\rm = 2 \log_a x - \log_a x$

$\rm = \log_a x$

d) $\rm a \log_a x = x$

Solution:
Given,

$\rm = a \log_a x$

e) $\rm \log_a a^x = x$

Solution:
Given,

$\rm = \log_a a^x$

$\rm = x \log_a a$

$\rm = x \times 1$

$\rm = x$

f) $\rm ( \log a)^2 - (\log b)^2 = \log(ab) \cdot \log(a/b)$

Solution:
Given,

$\rm ( \log a)^2 - ( \log b)^2$

Using formula of $\rm ( a^2 - b^2 ) = (a+b)(a-b)$

$\rm = (\log a + \log b) \times ( \log a - \log b)$

$\rm = \log (ab) \cdot log(a/b)$

g) $\rm \log(1 + 2 + 3) = \log 1 + \log 2 + \log 3$

Solution:
Given,

$\rm = \log (1 + 2 + 3)$

$\rm = \log(6)$

$\rm = \log (1 \times 2 \times 3)$

Using the product rule of logarithm

$\rm = \log 1 + \log 2 + \log 3$

h) $\rm x^{\log y - \log z} \cdot y ^{\log z - \log x} \cdot z ^{\log x - \log y} = 1$

Solution:
Given,

$\rm = x^{\log y - \log z} \cdot y ^{\log z - \log x} \cdot z ^{\log x - \log y}$


j) $\rm \log_a \sqrt{ a \sqrt{a \sqrt{a^2}}} = 1$

Solution:
Given,

$\rm = \log_a \sqrt{a \sqrt{a \sqrt{a^2}}}$

$\rm = \log_a \sqrt{a \sqrt {a \times a }}$

$\rm = \log _ a \sqrt{ a \sqrt{ a^2}}$

$\rm = \log _a \sqrt{a \times a }$

$\rm = \log _a \sqrt{a^2}$

$\rm = \log_a (a)$

$\rm = 1$

2) If $\rm a^2 + b^2 = 2ab$, show that $\log \frac{a+b}{2} = \frac{\log a + \log b}{2}$

Solution:
Given,


About the Textbook:

Name: Basic Mathematics Grade XI
Author(s): D.R. Bajracharya | R.M. Shresththa | M.B. Singh | Y.R. Sthapit | B.C. Bajracharya
Publisher: Sukunda Pustak Bhawan (Bhotahity, Kathmandu)
Telephone: 5320379, 5353537
Price: 695 /- (2078 BS)