Class 9 Advanced Maths Chapter 2 Logarithms Ex 2.1 Solutions

Students often refer to NCERT Class 9 Advanced Maths Solutions Chapter 2 Logarithms Ex 2.1 to verify their answers.

Advanced Maths Class 9 Exercise 2.1 Solutions

Class 9 Advanced Maths Ex 2.1 Solutions

Question 1.
Write an equivalent logarithmic statement for:
(a) 5³ = 125
Solution:
For exponential form b^x = a, the equivalent logarithmic form is given by
log_b a = x, where (b > 0, a > 0, b ≠ 1).
Here, b = 5, x = 3, a = 125.
Logarithmic form is log₅(125) = 3.

(b) (2)⁵ = 32
Solution:
For exponential form b^x = a, the equivalent logarithmic form is given by
log_ba = x, where (b > 0, a > 0, b ≠ 1).
Here, b = 2, x = 5, a = 32.
Logarithmic form is log₂(32) = 5.

(c) (7)^-1 = \(\frac{1}{7}\)
Solution:
For exponential form b^x = a, the equivalent logarithmic form is given by
log_ba = x, where (b > 0, a > 0, b ≠ 1).
Here, b = 7, x = -1, a = \(\frac{1}{7}\).
Logarithmic form is log₇ \((\frac{1}{7})\) = -1

(d) (3)^{\(-\frac{1}{2}\)} = \(\frac{1}{\sqrt{3}}\)
Solution: For exponential form b^x = a, the equivalent logarithmic form is given by
log_ba = x, where (b > 0, a > 0, b ≠ 1)
Here, b = 3, x = \(-\frac{1}{2}\), a = \(\frac{1}{\sqrt{3}}\)
Logarithmic form is log₃\(\left(\frac{1}{\sqrt{3}}\right)\) = \(\frac{1}{2}\)

Question 2.
Write an equivalent exponential statement for:
(a) log₂16 = 4
Solution:
For logarithmic form given by log_b a = x, where (b > 0, a > 0, b ≠ 1), the equivalent exponential form is b^x = a.
Here, b = 2, a = 16, x = 4.
Exponential form is 2⁴ = 16.

(b) log₉81 =2
Solution:
For logarithmic form given by log_b a = x, where (b > 0, a > 0, b ≠ 1), the equivalent exponential form is b^x = a. Here, b = 9, a = 81, x = 2.
Exponential form is 9² = 81.

(c) log₅\(\sqrt{5}\) = \(\frac{1}{2}\)
Solution: For logarithmic form given by log_ba = x, where (b > 0, a > 0, b ≠ 1), the equivalent exponential form is b^x = a.
Here, b = 5, a = \(\sqrt{5}\), x = \(\frac{1}{2}\).
Exponential form is 5^{\(\frac{1}{2}\)} = \(\sqrt{5}\).

(d) log₂\(\left(\frac{1}{2}\right)\) = -1
Solution:
For logarithmic form given by log_b = x, where (b > 0, a > 0, b ≠ 1), the equivalent exponential form is b^x = a.
Here, b = 2, a = \(\frac{1}{2}\), x = -1.
Exponential form is 2^-1 = \(\frac{1}{2}\).

Question 3.
Find the value of
(a) log₁₀ 1000
Solution:
Let log₁₀ 1000 = z.
Convert to exponential form first using:
b^y = x ⇔ log_b(x) = y.
∴ log₁₀ 1000 = z ⇔ 10^z = 1000 = 10³.
By comparing exponents, we get: ⇒ z = 3.
Substituting the value of z in the given expression, we get, log₁₀ 1000 = 3.

(b) log₆ 36
Solution:
Let log₆ 36 = z.
Convert to exponential form first using:
b^y = x ⇔ log^b (x) = y.
∴ log⁶ 36 = z ⇔ 6^z = 36 = 6².
By comparing exponents, we get: z = 2.
Substituting the value of z in the given expression, we get, log₆36 = 2.

(c) log₂ 64
Solution:
Let log₂ 64 = z.
Convert to exponential form first using:
b^y = x ⇔ log_b (x) = y.
∴ log₂ 64 = z ⇔ 2^z = 64 = 2⁶.
By comparing exponents, we get, z = 6.
Substituting the value of z in the given expression, we get, log₂ 64 = 6.