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) log_{25}5

Answer:

ii) log_{81}3

Answer:

iii) log_{2}(\(\frac{1}{16}\))

Answer:

iv) log_{7}1

Answer:

log_{7}1 = log_{7}7^{0} = 0 log_{7}7 = 0

v) log_{x}√x

Answer:

vi) log_{2}512

Answer:

log_{2}512 = log_{2}2^{9} [∵ 512 = 2^{9}]

= 9log_{2}2 [∵ log x^{m} = m log x]

= 9 × 1 [∵ log_{a}a = 1]

= 9

vii) log_{10}0.01

Answer:

viii) \(\log _{\frac{2}{3}}\left(\frac{8}{27}\right)\)

Answer:

ix) \(2^{2+\log _{2} 3}\)

Answer:

\(2^{2+\log _{2} 3}\) = 2^{2} . \(2^{\log _{2} 3}\) [∵ a^{m} . a^{n} = a^{m+n}]

= 4 × 3 [∵ \(\log _{\mathrm{a}} \mathrm{N}\) = N]

= 12

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) log_{2} 16 – log_{2} 2

Answer:

iii) 3 log_{64}4

Answer:

iv) 2 log 3 – 3 log 2

Answer:

2 log 3 – 3 log 2

= log 3^{2} – log 2^{3}

= 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 3^{2} – log 2

= log 10 + log 9 – log 2 [∵ m log a = log a^{m}]

= log \(\frac{10 \times 9}{2}\) [∵ log a + log b = log ab; log a – log b = log \(\frac{a}{b}\)]

= log 45

Question 3.

Evaluate each of the following in terms of x and y, if it is given x = log_{2}3 and y = log_{2} 5.

i) log_{2}15

Answer:

log_{2}15 = log_{2} 3 × 5

= log_{2}3 + log_{2}5 [∵ log mn = log m + log n]

= x + y

ii) log_{2}7.5

Answer:

iii) log_{2}60

Answer:

log_{2}60 = log_{2}2^{2} × 3 × 5

= log_{2}2^{2} + log_{2}3 + log_{2}5

= 2 log_{2}2 + x + y

= 2 + x + y

iv) log_{2}6750

Answer:

log_{2}6750

= log_{2}2 × 3^{3} × 5^{3}

= log_{2}2 + log_{2}33 + log_{2}5^{3}

= 1 + 3 log_{2}3 + 3 log_{2}5

= 1 + 3x + 3y

Question 4.

Expand the following,

i) log 1000

Answer:

log 1000 = log 10^{3}

= 3 log 10

= 3 × 1

= 3

ii) \(\log \left[\frac{128}{625}\right]\)

Answer:

iii) log x^{2}y^{3}z^{4}

Answer:

log x^{2}y^{3}z^{4} = logx^{2} + logy^{3} + logz^{4} [∵ log ab = log a + log b]

= 2 log x + 3 log y + 4 log z

[∵ log a^{m} = m log a]

iv) \(\log \frac{\mathbf{p}^{2} \mathbf{q}^{3}}{\mathbf{r}}\)

Answer:

iv) \(\log \sqrt{\frac{x^{3}}{y^{2}}}\)

Answer:

Question 5.

If x^{2} + y^{2} = 25xy, then prove that 2 log (x + y) = 3log3 + logx + logy.

Answer:

Given: x^{2} + y^{2} = 25xy

We know that (x + y)^{2} = x^{2} + y^{2} + 2xy

= 25xy + 2xy [∵ x^{2} + y^{2} = 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 3^{3} + 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:

(squaring on both sides)

⇒ (x + y)^{2} = (3√xy)^{2}

⇒ x^{2} + y^{2} + 2xy = 9xy

⇒ x^{2} + y^{2} = 9xy – 2xy = 7xy

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 = 10^{3}

Question 8.

If 2^{x+1} = 3^{1-x} then find the value of x.

Answer:

Given: 2^{x+1} = 3^{1-x}

log 2^{x+1} = log 3^{1-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

Question 9.

Is

i) log 2 is rational or irrational? Justify your answer.

Answer:

Let log_{10}2 = x

Then 10^{x} = 2

But 2 can’t be written as 10^{x} for any value of x

∴ log 2 is irrational.

ii) log 100 is rational or irrational? Justify your answer.

Answer:

Let log_{10}100 = x

⇒ log_{10}10^{2} = x

⇒ 2 log_{10}10 = x = 2

∴ log 100 is rational.

∴ log 100 = 2

Hence rational.