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

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:

ii) log₈₁3
Answer:

iii) log₂(\(\frac{1}{16}\))
Answer:

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

v) log_x√x
Answer:

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

vii) log₁₀0.01
Answer:

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

ix) \(2^{2+\log _{2} 3}\)
Answer:
\(2^{2+\log _{2} 3}\) = 2² . \(2^{\log _{2} 3}\)   [∵ a^m . aⁿ = 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) log₂ 16 – log₂ 2
Answer:

iii) 3 log₆₄4
Answer:

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 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 = 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:

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

iv) log₂6750
Answer:

log₂6750
= log₂2 × 3³ × 5³
= log₂2 + log₂33 + log₂5³
= 1 + 3 log₂3 + 3 log₂5
= 1 + 3x + 3y

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:

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 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 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:

(squaring on both sides)
⇒ (x + y)² = (3√xy)²
⇒ x² + y² + 2xy = 9xy
⇒ x² + y² = 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 = 10³

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 log₁₀2 = 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 log₁₀100 = x
⇒ log₁₀10² = x
⇒ 2 log₁₀10 = x = 2
∴ log 100 is rational.
∴ log 100 = 2
Hence rational.