Exercise 1

Calculate the value of y.

1    log _ {\frac {1}{2}} {0.25} = y

log _ {\sqrt {5}} {125} = y

3  log 0.001 = y

4 ln \frac {1} {e ^ {5}} = y

5 log_ {\sqrt {3}}  \sqrt [5] {\frac {1}{81}} = y

Superprof

Exercise 2

Apply the definition of logarithms and calculate the value of x:

1 log _ 2 {32} = x

2 log _9 \frac{1}{3} = x

3 log _{\frac {1}{2}} {0.25} = x

4 log _ 9 \sqrt[4] {3} = x

5 log _ {\sqrt {2}} {\frac {1}{4}} = x

6 log _ x 81 = -4

7 log_2 x ^ 3 = 6

Exercise 3

Knowing that log 2 = 0.3010, calculate the following logarithms.

log 0.02

2 log \sqrt [4] {8}

3 log 5

4 log 0.0625

Exercise 4

Calculate:

1   ln \frac {x ^ 2 \cdot y \cdot (m + n)} {m \cdot n}

log _ 2 {\frac{a ^2 - b^2} {a \cdot b}}

log \sqrt [2] {2 \sqrt {2} \sqrt{2}}

Exercise 5

Using logarithms, calculate the value of x.

1 x =\sqrt [5] {493}

2 x = \frac {\sqrt [3] {0.3688}} { 22.958 ^ 5}

3 x = \frac {425 \cdot \sqrt {2.73}} {\sqrt [3] {48.4}}

Solution of exercise 1

Calculate the value of y.

1    log _{\frac {1}{2}} {0.25} = y

Covert it into the exponential form like this:

\frac {1}{2} ^ y = 0.25

Since, 0.25 is equal to the square of \frac{1}{2}, hence \frac {1} {2} ^ y = \frac {1} {2} ^ 2

So the final answer is y = 2

 

2      log _ {\sqrt {5}} {125} = y

Convert it into the exponential form like this:

\sqrt{5} ^ y = 125

5 ^ \frac{1}{2} = 5 ^ 3

Hence, y = 6

 

3      log 0.001 = y

Convert it into the exponential form. Since it has no base, so we will assume that it is a common algorithm with base 10:

10 ^ y = 0.001

0.001 can be written as \frac{1}{1000} or \frac{1}{10 ^ 3}. According to the negative exponent rule, we can move the base to the numerator and the sign of 3 will change from positive to negative.

10 ^ y = 10 ^ {-3}

Hence, y = -3

 

4      ln \frac {1} {e ^ {5}} = y

Convert it into the exponential form like this:

e ^ y = \frac {1} {e ^ 5}

Apply the negative exponent rule, to write the right hand side of the equation in numerator:

e ^ y = e ^ {-5}

Hence, y = -5

 

5       log_ {\sqrt {3}}  \sqrt [5] {\frac {1}{81}} = y
Convert the above logarithmic form into the exponential function like this:
\sqrt {3} ^ y = \sqrt [5] {\frac {1}{81}}
The radicals can be written in the form of fractions like this:3 ^ {\frac{1}{2}} = 3 ^ \frac {-4}{5}Multiply the exponents of both sides by 2 to get the value of yHence, y = \frac {-8}{5}

Solution of exercise 2

Apply the definition of logarithms and calculate the value of x:

1   log _2 {32} = x

Convert the above logarithmic equation into the exponential form like this:

2 ^ x = 32

Since 2 raised to the power 5 is equal to 32, so we can write the equation as:

2 ^ x = 2 ^ 5

x= 5

 

2    log _9 \frac{1}{3} = x

Convert the above logarithmic equation into the exponential form like this:

9 ^ x = \frac{1}{3}

Since 3 raised to the power 2 is equal to 9 and the fraction is equal to the negative power in the numerator, so we can write the expression as:

3 ^ {2x} = 3 ^ {-1}

x = -\frac {1}{2}

 

3    log _{\frac {1}{2}} {0.25} = x

Convert the equation into exponential form like this:

(\frac{1}{2} ^ x = (\frac{1}{2}) ^ 2

Hence, x = 2

 

log _ 9 \sqrt[4] {3} = x

Convert the logarithmic form into an exponential form like this:

9 ^ x = \sqrt[4] {3}

Since 3 raised to the power 2 is equal to 9 and the radical can be written in fractional exponential form, so the expression will be:

3 ^ {2x} = 3 ^ \frac {1}{4}

Hence, x = \frac{1}{8}

 

5    log _ {\sqrt {2}} {\frac {1}{4}} = x

Convert the above logarithmic equation into the exponential form like this:

(\sqrt {2}) ^ x = \frac {1}{4}

The radical form can be written in fractional exponential form like this:

2 ^{\frac{1}{2} x} = 2 ^ {-2}

\frac {1}{2}x = -2

Multiply 2 on both the sides of the equation to get the final answer:

x = -4

 

6    log _ x 81 = -4

Convert the above logarithmic equation into an exponential form like this:

x ^ {-4} = 81

It can be written into the fractional form like this:

x ^ 4 = \frac {1}{81}

x ^ 4 = \frac {1}{3} ^ 4

Hence, x = 4

 

7    log_2 x ^ 3 = 6

It can be written in exponential form like this:

x ^ 3 = 2 ^ 6

If 2 ^ 6 = 64, then 4 ^ 3 = 64.

Hence, x = 4

 

Solution of exercise 3

Knowing that log 2 = 0.3010, calculate the following logarithms.

1    log 0.02

Since 0.02 can be written in the fractional form \frac{2}{100}, hence we will write the above logarithmic function as:

log (\frac {2}{100}

Apply the logarithm quotient rule here:

log 2 - log 10 ^ 2

Apply  logarithm power rule here:

log 2 - 2 log 10

Since, log 10 = 1, so the resultant expression will be written like this:

log 2 - 2 = 0.3010 - 2 = -1.699

 

2      log \sqrt [4] {8}

Applying the logarithm root rule here we get the following expression:

\frac {3}{4} \cdot 0.3010 = 0.2257

 

3     log 5

It can be written as:

log \frac{10}{2}

Applying the logarithm quotient rule here, we will get the following expression:

= log 10 - log 2

= 1 - 0.3010 =0.699

 

4    log 0.0625

The above logarithm can be written in the fractional form like this:

log \frac {625}{10000}

625 = 5 ^ 4 and 10000 can be written as a product of 2 and 4 raised to the power 4 like this:

=log ({\frac 5 ^ 4}{2 ^ 4 \cdot 5 ^ 4}

= log \frac {1}{2 ^ 4}

Apply the logarithm quotient rule here to get the following expression:

log 1 - log 2 ^ 4

Since, log 1 =0 and the root rule can be applied to log 2 ^ 4 , so we can write the expression as:

0 = 4 log 2 = -1.2040

 

Solution of exercise 4

Calculate:

1   ln \frac {x ^ 2 \cdot y \cdot (m + n)} {m \cdot n}

Apply the logarithm quotient rule here to get the following expression:

= ln [x ^ 2 \cdot y \cdot (m + n)] - ln(m \cdot n)

= ln x ^ 2 +lny +ln(m +n) - (ln m + ln n)

Simplify the expression and apply logarithm power rule here:

=2 ln x +ln y + ln (m + n) - ln m - ln n

 

2       log _ 2 {\frac{a ^2 - b^2} {a \cdot b}}

The formula in the numerator can be written as the product of two factors like this:

log _2 \frac {(a + b) \cdot (a - b) } {a \cdot b}

Applying the logarithm quotient rule here, we will get the following expression:

= log _ 2 [ ( a + b) \cdot (a - b)] -log _ 2 (a \cdot b)

= log _ 2 [ ( a + b) \cdot (a - b)] -log _ 2 a +log_ 2 b

 

3         log \sqrt [2] {2 \sqrt {2} \sqrt{2}}

The above expression can be written as follows after applying the logarithm product rule:

= log 2 +  <span class="numero_v">log \sqrt {2 \sqrt {2} \sqrt{2}}

= log 2 + \frac {1}{2} [log 2 + \frac{1}{2} log (2 \sqrt {2})]

= log 2 + \frac {1}{2} [log 2 +\frac{1}{2} (log 2 + log \sqrt{2})]

=log 2 +\frac{1}{2} [ log 2 +\frac{1}{2} (log 2 + \frac{1}{2} log 2)]

=log 2 + \frac{1}{2} log 2 + \frac {1}{4} log 2 + \frac {1}{8} log 2 = \frac {15}{8} log 2

 

Solution of exercise 5

Remember that if you are asked to write the answers in numbers, then you can use your calculators to compute the final value of the variable. On the other hand, if you are asked to simplify the expression, you can simply apply the logarithm rules and write the simplified form of the expression.

In these exercises, we are asked to compute the value of x, so we have to write the answer in digits.

Using logarithms, calculate the value of x.

1         x =\sqrt [5] {493}

Take log on both sides of the equation. The resultant equation will be like this:

log x = log \sqrt [5] {493}

log x = \frac {1}{5} \cdot log 493 = \frac {1}{5} \cdot 2.6928 = 0.5386

Calculate the antilogarithm.

x = antilog 0.5386 =3.456

2      x = \frac {\sqrt [3] {0.3688}} { 22.958 ^ 5}

Take log on both sides of the equation to get the following expression:

log x = log \frac{sqrt[3] {0.3688}} {22.958 ^ 5}

Apply the logarithm quotient rule here:

log x = log \sqrt [3] {0.3688} - log 22.958 ^ 5 = \frac {1}{3} log 0.3688 - 5 log 22.958

log x = -6.949

Take anti -log here:

x = antilog -6.949 = 1.124 \cdot 10^ -7

3        x = \frac {425 \cdot \sqrt {2.73}} {\sqrt [3] {48.4}}

Take log on both sides of the equation to get the following expression:

log x = log \frac {425 \cdot \sqrt {2.73}} { \sqrt [3] {48.4}}

Apply the logarithm quotient rule here to get the following expression:

log x = log(425 \cdot \sqrt {2.73} - log \sqrt [3] {48.4}

log x = 2.6284 + \frac {1}{2} \cdot 0.4362 - \frac {1}{3} \cdot 1.6848 = 2.2 849

Take anti log here to isolate the variable x on the left hand side of the equation:

x = antilog 2.2849 = 192.71

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.

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