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Solve Problems: 1) Find the logarithm of 1/256 to the base 2√2.

16 13/5 -16/3 12 View Answer

Answer: C

Explanation:

Let log_{2√2} [1/256] = x

We know that log_{a} y = x is similar to a^{x} = y

So, we can write it as [1/256] = (2√2) ^{x}

Or, (2√2) ^{x} = [1/2^{8} ]

Or, [2^{1} * 2^{1/2} ]^{x} = 1/2^{8}

Or, 2^{3x/2} = 2-8

Therefore, 3x/2 = -8

Hence, x = (-8 * 2)/ 3 = -16/3

2) If log_{a} [1/36] = -2/3, find the value of a.

6 8 9 216 View Answer

Answer: D

Explanation:

ATQ, log_{a} [1/36] = -2/3

We know that log_{a} y = x is similar to a^{x} = y

So, a ^{(-2/3) } = 1/36

Or, a ^{(-2/3) } = 1/6^{2}

Or, a^{ (-2/3) } = 6^{-2}

Now, multiply and divide by 3 in the power of 6 to make the power equals to power of a.

So, a ^{(-2/3) } = 6^{3(-2/3)}

On comparing both side, we get a = 6^{3}

Therefore, a = 216

3) Find the value of x

Log_{4} (log_{8} 64) = log_{5} x

2 6 5 √5 View Answer

Answer: D

Explanation:

ATQ, Log_{4} (log_{8} 64) = log_{5} x…… (i)

Let, log_{8} 64 = a

Or, 64 = 8^{a} , or 8^{a} = 8^{2}

That means a=2

Now, put log_{8} 64 = 2in equation 1.

Log_{4} (2) = log_{5} x…… (ii)

Now, let log_{4} 2 = s

Or, 4^{s} = 2, or 2^{2S} = 2

Now, on comparing both side we get 2_{s} = 1, or s = ½

Put the value of s in equation 2

So, log_{5} x = ½

Therefore, x = 5^{1/2} = √5

4) The equation, a^{2} + b^{2} = 7ab equals to

View Answer

Answer: A

Explanation:

Here, a^{2} + b^{2} = 7ab

Add both sides 2ab to make a formula.

Or, a^{2} + b^{2} + 2ab = 7ab + 2ab

Or, (a+b) ^{2} = 9ab

Or, (a+b) ^{2} / 9 = ab

Or, [1/3 (a+b)]^{2} = ab

Now, taking log both sides

Therefore, log [1/3 (a+b)]^{2} = log ab

We know that log m*n = log m + log n, and log m^{n} = n log m

So, 2 log [1/3 (a+b)] = log a + log b

Therefore, log [1/3 (a+b)] = 1/2(log a + log b)

= a^{2} + b^{2}

= 7ab

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5) If (log_{3} x)(log_{x} 2x)(log_{2x} y) = log_{x} x^{2} , find y.

6 9 5 7 6) Add the equation x = 1 + log_{a} bc, y = 1 + log_{b} ca and z = 1 + log_{c} ab. Find the value of

xyz = xy + yz + zx X^{2} yz = xyz + yx + 1 (xz + 1)^{2} = xy – yz – zx (xz + y) = xy + yz + zx 7) Find the characteristics of the logarithms of i) 5631, ii) 5.678, iii) 56.23.

3, 1, 5 3, 0, 1 6, 5, 9 8, 7, 6 View Answer

Answer: B

Explanation:

i) The number 5631> 1 and the number of digits in the integral part are 4. Therefore, the characteristics of the logarithm are 4-1 = 3,

ii) In the number 5.678, the number of digits in the integral part = 1. Therefore, characteristics of the logarithm are 1-1 =0

iii) In this case, the number of digits in an integral part of the number 56.23 are 2. Therefore, the characteristics of the logarithm = 2-1 = 1

So, the option b is correct.

8) If log 2 = 0.30103, log 3 = 0.47712, the number of digits in 6^{20} is

8 12 16 20 View Answer

Answer: C

Explanation:

We have to take 6^{20} with log We know that log m^{n} = n log m Now, log 6^{20} = 20 log 6 Or, 20 log (2*3) We know that log (m*n) = log m + log n Now, 20 [log 2 + log 3] = 20 [0.30103 + 0.47712] = 20 [0.77815] Since, the number of digits in 6^{20} = 15.563, round up value = 16 Therefore the number of digits = 16

9) The logarithm of 0.0001 to the base 0.001 is equal to

4/3 5/3 7/3 2/3 View Answer

Answer: A

Explanation:

Let Log_{0.001} (0.0001) = x

We know that log_{x} y = a equals to x^{a} = y.

So, (0.001)^{x} = 0.0001 Or, = [1/10000]

Or, =

Therefore, compare both side, we get 3x = 4.

Hence, x = 4/3

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10) If log_{b} x = ͞5.1342618, then the value of log_{10} (x^{(1/4)} ) will be

͞1.2835655 ͞2.7164345 ͞2.7835655 ͞3.2164345 View Answer

Answer: C

Explanation:

Here log_{b} x = ͞5.1342618 is given and log_{10} (x^{(1/4)} ) is asked

That means here b also treat as base 10

Now, we can say log_{10} x = ͞5.1342618 = -5 + 0.1342618 => -4.8657382

Therefore, log_{10} (x^{1/4} ) = ¼ log_{10} x

Or, ¼ (-4.8657382) = -1.21643455, but it is not in the option.

To make it -2, we have to subtract 0.7835655 from -1.21643455

i.e., -1.21643455 – 0.7835655 = -2

Now, to get back on the original value we have to add 0.7835655 in -2.

Or, -2 + 0.7835655 = ͞2.7835655

Hence, the c is correct.