WBBSE Solutions For Class 9 Maths Algebra Chapter 6 Logarithm

Algebra Chapter 6 Logarithm

⇒ If a^x = N (N > 0, a > 0, N ≠ 1), then x is called the logarithm of the number N to the base a and is written as x = log_aN

⇒ Thus, if a^x = N, then x = log_aN

⇒ Conversely, if x = log_aN, then a^x = N

Formulae on logarithm: (M > 0, N > 0, a > 0, b > 0, a ≠ 1, b ≠ 1 and n is any real number)

1. \(\log _a(\mathrm{MN})=\log _a \mathrm{M}+\log _a \mathrm{~N}\)
2. \(\log _a\left(\frac{M}{N}\right)=\log _a \mathrm{M}-\log _a \mathrm{~N}\)
3. \(\log _a \mathrm{M}^n=x \log _a \mathrm{M}\)
4. \(\log _a \mathrm{M}=\log _b \mathrm{M} \times \log _a b\)
5. \(\log _a 1=0\)
6. \(\log _a a=1\)
7. \(a^{\log _a M}=M\)
8. \(\log _b a \times \log _a^b\) = 1
9. \(\log _b a=\frac{1}{\log _a b}\)
10. \(\log _b M=\frac{\log _a M}{\log _a b}\)

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Algebra Chapter 6 Logarithm True Or False

Example 1. log_a1 = 1

Solution: The statement is False.

Example 2. log_aa = 1

Solution: The statement is True.

Example 3. a^loga^M = a

Solution: The statement is False.

Example 4. \(\log _b a=\frac{1}{\log _a b}\)

Solution: The statement is True.

Example 5. \(\log _a \frac{1}{a}\) = 1

Solution: The statement is False.

Example 6. \(\log _b a \times \log _a b\) = 1

Solution: The statement is True.

Example 7. Solving the equation 2^x = 7, we get x = log₂⁷.

Solution: The statement is True.

Example 8. Logarithm is said that idea of logarithm is 6th fundamental process.

Solution: The statement is False.

Example 9. log_aM is undefined when a = 1.

Solution: The statement is True.

Example 10. log_aM is defined when a = 0.

Solution: The statement is False.

Algebra Chapter 6 Logarithm Fill In The Blanks

Example 1. If log₁₀x + 1 = 0 then x _______

Solution: \(\frac{1}{10}\)

⇒ log₁₀x = -1

⇒ x = 10-1 = \(\frac{1}{10}\)

Example 2. If b² = ae, then log_b(abc) = _______

Solution: 3

⇒ \(\log _b(b . a c)=\log _b\left(b . b^2\right)=\log _b b^3=3 \log _b b=3\)

Example 3. If \(\log _{a b} a=x\), then \(\log _{a b} b\) = ______

Solution: 1 – x

⇒ \(\log _{a b} a+\log _{a b} b=\log _{a b}(a b)=1\)

∴ \( x + \log _{a b} b=1\)

⇒ or, \(\log _{a b} b=1-x\)

Example 4. If log₄(3x+4)= 3 then the value of x is

Solution: 20

⇒ 4³ = 3x + 4

⇒ x = \(\frac{64-4}{3}\) = 20

Example 5. If log₂₀2 = a then the value of log₂₀10 is ______

Solution: 1 – a.

⇒ \(\log _{20} 10=\log _{20} \frac{20}{2}\)

= \(\log _{20} 20-\log _{20} 2=1-a\)

Example 6. The value of log₁₂(log₁₉16 x log₁₆19) = _______

Solution: 0

⇒ log₁₂(1) = 0

Example 7. If log4 = -2, then n = _______

Solution: \(\frac{1}{2}\)

⇒ \(x^{-2}=4\)

∴ \(\left(\frac{1}{x}\right)^2=2^2\)

∴ \(x=\frac{1}{2}\)

Example 8. If log₁₀(x – 2) = 1 – log₁₀2 then x = _____

Solution: 7

⇒ log₁₀ (x -2)+ log₁₀2 = 1

⇒ or, log₁₀(2x – 4) = 1

∴ 2x – 4 = 4

⇒ x = 7

Example 9. The value of logs log_√28 = ______

Solution: \(2^{\log 2 x^2 x^2}\)

⇒ \(\log _6 \cdot 3 \cdot \log _{\sqrt{2}} 2=\log _6 3 \cdot 2 \times 2=\log _6 6=1\)

Example 10. The value of \(4 \log 2^x\) is _______

Solution: \(_2 2 \log _2 x=_2 \log _2 x^2 x^2\)

Algebra Chapter 6 Logarithm Short Answer Type Questions

Example 1. Let us calculate the value of log₄ log₄ log₄ 256.

Solution: log₄ log₄ log₄(4)⁴ = log₄ log₄4

= log₄1 = 0

Example 2. Calculate the value of \(\frac{a^n}{b^n}+\log \frac{b^n}{c^n}+\log \frac{c^n}{a^n}\)

Solution: log a^x – log b^x + log b^x – log c^x + log c^x – log a^x = 0

Example 3. Show that \(a^{\log _a x}=x\)

Solution: Let \(\log _a x=\mathrm{M}\)

∴ \(a^M=x\)

Example 4. If \(\log _e 2 \log _x 25=\log _{10} 16 . \log _e 10\)

Solution: \(\log _e 2 \log _x 5^2=\log _{10} 2^4 \log _e 10\)

⇒ or, \(\log _e 2 . \log _x 5^2=4 \log _{10} 2 \log _e 10\)

⇒ or, \(\log _e 2 \log _x 5^2=4 \log _e 2\)

∴ \(x^4=5^2\)

∴ \(x^4=100\),

∴ \(x= \pm \sqrt{5}\)

Example 5. Find the value of \(\log _{2 \sqrt{3}} 1728\).

Solution: let \(\log _{2 \sqrt{3}} 1728\)

∴ \((2 \sqrt{3})^x=1728\)

or, \((2 \sqrt{3})^x=2^6 3^3=(2 \sqrt{3})^6\)

∴ x = 6

Example 6. Expression terms of \(N: \frac{1}{2} \log _3 M+\log _3 N=1\)

Solution: \(\log _3 \sqrt{M}+\log _3 N^3=1\)

⇒ or, \(\log _3 \sqrt{M} N^3=1\)

⇒ or, \(\sqrt{M} N^3=3\)

⇒ or, \(M=\left(\frac{3}{N^3}\right)^2=\frac{9}{N^6}\)

Example 7. Prove that (log x)² – (log y)² = \(\log (x y) \log \frac{x}{y}\)

Solution: (log x + log y) (log x log y) = log (xy) log\(\frac{x}{y}\)

Example 8. If \(\log _{30} 3=a \text { and } \log _{30} 5=b\), then find the value of \(\log _{30} 8\)

Solution: \(\log _{30} 2^3=3 \log _{30} \frac{30}{15}=3\left(\log _{30} 30-\log _{30} 15\right)\)

= \(3\left\{1-\log _{30}(3 \times 5)\right\}=3(1-a-b)\)

Example 9. Find the base when 3 is the logarithm of 343.

Solution: log_x 343 = 3

∴ x³ = 343 = 7³

∴ x = 7

Example 10. Find the simplest value of \(\log _3 5 \times \log _{25} 27\)

Solution: \(\log _3 5 \times \log _{5^2} 3^3=\frac{3}{2} \log _3 5 \times \log _5 3=\frac{3}{2}\)