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How do you solve ${0.25^x} - 0.5 = 2$?

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Last updated date: 27th Feb 2024
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IVSAT 2024
Answer
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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,
$\log {x^y} = y\log x$