Answer

Verified

387.3k+ views

**Hint:**First, move the constant part on one side. After that take log on both sides and apply the property of the log, $\log {a^b} = b\log a$. Then change the decimal value in the fraction part. Then, apply the property of log, $\log \dfrac{a}{b} = \log a - \log b$. After that, divide both sides by the coefficients of $x$ to get the desired result.

**Complete step by step solution:**

Let us understand the definition of log first.

Logarithms are the opposite of exponentials, just as the opposite of addition is subtraction and the opposite of multiplication is division.

In other words, a logarithm is essentially an exponent that is written in a particular manner.

Logarithms can make multiplication and division of large numbers easier, because adding logarithms is the same as multiplying, and subtracting logarithms is the same as dividing.

The given expression is ${0.25^x} - 0.5 = 2$.

Move the constant part on the right side of the expression,

$ \Rightarrow {0.25^x} = 2.5$

Take a log on both sides of the expression.

$ \Rightarrow \log {0.25^x} = \log 2.5$

We know that the power law of log is,

$\log {a^b} = a\log b$

Using the above law, the expression will be,

$ \Rightarrow x\log 0.25 = \log 2.5$

Change the decimal part in the fraction part,

$ \Rightarrow x\log \dfrac{{25}}{{100}} = \log \dfrac{{25}}{{10}}$

Cancel out the common factors,

$ \Rightarrow x\log \dfrac{1}{4} = \log \dfrac{5}{2}$

We know that,

$\log \dfrac{a}{b} = \log a - \log b$

Using the above law, the expression will be,

$ \Rightarrow x\left( {\log 1 - \log 4} \right) = \log 5 - \log 2$

We know that, $\log 1 = 0$. Then,

$ \Rightarrow - x\log 4 = \log 5 - \log 2$

Now, divide both sides by $ - \log 4$ to get the value of $x$,

$ \Rightarrow x = \dfrac{{\log 5 - \log 2}}{{ - \log 4}}$

Multiply numerator and denominator by $ - 1$ and simplify,

$\therefore x = \dfrac{{\log 2 - \log 5}}{{\log 4}}$

**Hence, the value of x is $\dfrac{{\log 2 - \log 5}}{{\log 4}}$.**

**Note:**A logarithm with base 10 is a common logarithm. In our number system, there are ten bases and ten digits from 0-9, here the place value is determined by groups of ten. You can remember common logarithms with the one whose base is common as 10.

Change of base rule law,

${\log _y}x = \dfrac{{\log x}}{{\log y}}$

Product rule law,

$\log xy = \log x + \log y$

Quotient rule law,

$\log \dfrac{x}{y} = \log x - \log y$

Power rule law,

$\log {x^y} = y\log x$

Recently Updated Pages

What number is 20 of 400 class 8 maths CBSE

Which one of the following numbers is completely divisible class 8 maths CBSE

What number is 78 of 50 A 32 B 35 C 36 D 39 E 41 class 8 maths CBSE

How many integers are there between 10 and 2 and how class 8 maths CBSE

The 3 is what percent of 12 class 8 maths CBSE

Find the circumference of the circle having radius class 8 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

One cusec is equal to how many liters class 8 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Change the following sentences into negative and interrogative class 10 english CBSE