AP State Board Syllabus AP SSC 10th Class Maths Textbook Solutions Chapter 1 Real Numbers Ex 1.5 Textbook Questions and Answers.

AP State Syllabus SSC 10th Class Maths Solutions 1st Lesson Real Numbers Exercise 1.5

10th Class Maths 1st Lesson Real Numbers Ex 1.5 Textbook Questions and Answers

Question 1.
Determine the values of the following,
i) log255
Answer:
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 1

AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5

ii) log813
Answer:
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 2
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 3

iii) log2(\(\frac{1}{16}\))
Answer:
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 4

iv) log71
Answer:
log71 = log770 = 0 log77 = 0

AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5

v) logx√x
Answer:
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 5

vi) log2512
Answer:
log2512 = log229    [∵ 512 = 29]
= 9log22   [∵ log xm = m log x]
= 9 × 1    [∵ logaa = 1]
= 9

vii) log100.01
Answer:
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 6

viii) \(\log _{\frac{2}{3}}\left(\frac{8}{27}\right)\)
Answer:
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 7

ix) \(2^{2+\log _{2} 3}\)
Answer:
\(2^{2+\log _{2} 3}\) = 22 . \(2^{\log _{2} 3}\)   [∵ am . an = am+n]
= 4 × 3 [∵ \(\log _{\mathrm{a}} \mathrm{N}\) = N]
= 12

AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5

Question 2.
Write the following expressions as log N and find their values.
i) log 2 + log 5
Answer:
log 2 + log 5
= log 2 × 5   [∵ log m + log n = log mn]
= log 10
= 1

ii) log2 16 – log2 2
Answer:
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 8

iii) 3 log644
Answer:
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 9

iv) 2 log 3 – 3 log 2
Answer:
2 log 3 – 3 log 2
= log 32 – log 23
= log 9 – log 8
= log \(\frac{9}{8}\)

v) log 10 + 2 log 3 – log 2
Answer:
log 10 + 2 log 3 – log 2
= log 10 + log 32 – log 2
= log 10 + log 9 – log 2    [∵ m log a = log am]
= log \(\frac{10 \times 9}{2}\)    [∵ log a + log b = log ab; log a – log b = log \(\frac{a}{b}\)]
= log 45

AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5

Question 3.
Evaluate each of the following in terms of x and y, if it is given x = log23 and y = log2 5.
i) log215
Answer:
log215 = log2 3 × 5
= log23 + log25   [∵ log mn = log m + log n]
= x + y

ii) log27.5
Answer:
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 10

iii) log260
Answer:
log260 = log222 × 3 × 5
= log222 + log23 + log25
= 2 log22 + x + y
= 2 + x + y

iv) log26750
Answer:
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 11
log26750
= log22 × 33 × 53
= log22 + log233 + log253
= 1 + 3 log23 + 3 log25
= 1 + 3x + 3y

AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5

Question 4.
Expand the following,
i) log 1000
Answer:
log 1000 = log 103
= 3 log 10
= 3 × 1
= 3

ii) \(\log \left[\frac{128}{625}\right]\)
Answer:
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 12

iii) log x2y3z4
Answer:
log x2y3z4 = logx2 + logy3 + logz4 [∵ log ab = log a + log b]
= 2 log x + 3 log y + 4 log z
[∵ log am = m log a]

iv) \(\log \frac{\mathbf{p}^{2} \mathbf{q}^{3}}{\mathbf{r}}\)
Answer:
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 13

iv) \(\log \sqrt{\frac{x^{3}}{y^{2}}}\)
Answer:
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 14

AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5

Question 5.
If x2 + y2 = 25xy, then prove that 2 log (x + y) = 3log3 + logx + logy.
Answer:
Given: x2 + y2 = 25xy
We know that (x + y)2 = x2 + y2 + 2xy
= 25xy + 2xy    [∵ x2 + y2 = 25xy given]
(x + y)2 = 27xy
Taking ‘log’ on both sides
log (x + y)2 = log 27xy
2 log (x + y) = log 27 + log x + log y
= log 33 + log x + log y
⇒ 2 log (x + y) = 3log3 + log x + log y

Question 6.
If \(\log \left(\frac{\mathbf{x}+\mathbf{y}}{3}\right)\) = \(\frac{1}{2}\) (log x + log y), then find the value of \(\frac{x}{y}+\frac{y}{x}\).
Answer:
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 15
(squaring on both sides)
⇒ (x + y)2 = (3√xy)2
⇒ x2 + y2 + 2xy = 9xy
⇒ x2 + y2 = 9xy – 2xy = 7xy
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 16

Question 7.
If (2.3)x = (0.23)y = 1000 then find the value of \(\frac{1}{x}-\frac{1}{y}\).
Answer:
Given (2.3)x = (0.23)y = 1000 = 103
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 17

AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5

Question 8.
If 2x+1 = 31-x then find the value of x.
Answer:
Given: 2x+1 = 31-x
log 2x+1 = log 31-x
(x + 1) log 2 = (1 – x) log 3
x log 2 + log 2 = log 3 – x log 3
x log 2 + x log 3 = log 3 – log 2
x (log 3 + log 2) = log 3 – log 2
AP SSC 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 18

Question 9.
Is
i) log 2 is rational or irrational? Justify your answer.
Answer:
Let log102 = x
Then 10x = 2
But 2 can’t be written as 10x for any value of x
∴ log 2 is irrational.

ii) log 100 is rational or irrational? Justify your answer.
Answer:
Let log10100 = x
⇒ log10102 = x
⇒ 2 log1010 = x = 2
∴ log 100 is rational.
∴ log 100 = 2
Hence rational.