Logarithm

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Understanding Log: a^x=b [here a is called base, x means exponent and b is the result]
We can say the above expression as: log_ab=x
Hence we can say that logarithm is an equivalent way to express an exponential identity and the above two expressions are same.
Generally, the base is taken as 10 when it is not written as subscript. Therefore,
log b means log₁₀b, if base is not given and we assume it 10.

If log₃x = 2. Find the value of x.

View Answer

log₃x = 2
⇒ 3² = x
∴ x = 9

If log₃[log₅(log₂a)] = 0, find the value of a

None of the above

View Answer

log₃[log₅(log₂a)] = 0
log₃[log₅(log₂a)] = log₃1 [As log₃1 = 0]
or, log₅(log₂a) = 1
or, 5¹ = log₂a
or, log₂a = 5
or, 2⁵ = a
∴ a = 32

If log₄64=b, find the value of b.

-3

View Answer

log₄64 = b
or, b = log₄64
or, 64 = 4^b
or, 4³ = 4^b
∴ b = 3

What is the value of log₃

(

1 / 9

)

-2

-1

View Answer

Let, log₃

(

1 / 9

)

= a
or,

1 / 9

= 3^a
or,

1 / 3²

= 3^a
or, 3⁻² = 3^a
∴ a = −2

if log _2√2

(

1 / 64

)

= a, find the value of a.

-16

-8

-4

View Answer

1 / 64

= (2√2)^a

1 / 64

= 2^(3a/2)
or,

1 / 2⁶

= 2^(3a/2)
or, 2⁻⁶ = 2^(3a/2)
or, −6 =

3a / 2

or, −6 x 2 = 3a
or, a =

6 x 2 / 3

= −2 x 2 = −4

If log_x∛3 =

1 / 15

, What is the value of x ?

243

View Answer

log_x∛3 =

1 / 15

or, 3√3 = x^1/15
or, x^1/15 = 3√3
or, x = (3)^{(1/3) x 15}
or, x = (3)⁵
∴ x = 243

If log_b√3 =

1 / 8

, find the value of b.

View Answer

log_b√3 =

1 / 8

or, √3 = b^1/8
or, b^1/8 = √3
or, b = (3)^{(1/2) x 8}
or, b = (3)⁴
∴ b = 81

Find the value of: 3 ^log₃5

View Answer

Shortcut:
a^log_an = n
Let z = a^log_an
Presenting the above exponential identity in logarithm
log_a(z) = log_a(n)
∴ z = n
or, a^log_an = n
Using the shortcut, we get:
3^log₃5 = 5
Hence, its value is 5.

Find the value of 2^{2 − log₂5}

1 / 5

4 / 5

View Answer

Shortcut:
a^log_an = n
Let z = a^log_an
Presenting the above exponential identity in logarithm
log_a(z) = log_a(n)
∴ z = n
or, a^log_an = n
Here, 2^{2 − log₂5}
or, 2² x 2^{− log₂5}
or, 2² x 2^{log₂5⁽⁻¹⁾}
or, 2² x 5⁻¹
or,

2 x 2 / 5

or,

4 / 5

10.

Find the value of 5^{3 + log₅5 − log₂₅5}

125

125√5

None of the above

View Answer

= 5³ x 5^log₅5 x 5^{−log₂₅5}
= 5³ x 5¹ x 5^{−log₂₅5}
= 5³ x 5¹ x 5^−log_5²5
= 5³ x 5 x 5^{−1/2log₅5}
= 5³ x 5 x (5^log₅5)^−1/2
= 5³ x 5 x (5)^−1/2
= 5³ x 5 x

1 / √5

= 5³ x 5 x

√5 / √5 x √5

= 5³ x 5 x

√5 / 5

= 5³ x √5
= 125√5

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