Question

How do you solve ${{\log }_{x}}25$ = -0.5?

Answer 1

Hint: Solving logarithmic expressions is quite easy. Here in this problem, we have to find the value of ‘x’. You should know some properties which will be used to evaluate such expressions.
Logarithmic function is denoted as: ${{\log }_{b}}x=n$ or${{b}^{n}}=x$
Power rule: $a\log x=\log {{x}^{a}}$
$\Rightarrow {{x}^{{{\log }_{x}}\left( a \right)}}=a$
Multiplicative inverse of ${{a}^{-x}}=\dfrac{1}{{{a}^{x}}}$ or ${{a}^{x}}=\dfrac{1}{{{a}^{-x}}}$

Complete step by step answer:
Now, let’s begin to solve the question.
Logarithm is nothing but the power to which number must be raised to get some other values. It is the most convenient way to express large numbers.
Logarithmic function is denoted as: ${{\log }_{b}}x=n$ or${{b}^{n}}=x$
There are basically two types of logarithm: common logarithm and natural logarithm.
Mostly we use common logarithm. It is denoted as log base 10 or simply log. Whereas natural log is denoted by the natural base i.e. ‘e’ and represented as ln or loge.
Let’s see some rules to solve logarithms. They are as follows:
Product rule: ${{\log }_{b}}\left( mn \right)={{\log }_{b}}m+{{\log }_{b}}n$
Quotient rule: ${{\log }_{b}}\left( \dfrac{m}{n} \right)={{\log }_{b}}m-{{\log }_{b}}n$
Power rule: ${{\log }_{b}}\left( {{m}^{n}} \right)=n{{\log }_{b}}m$
$\Rightarrow {{x}^{{{\log }_{x}}\left( a \right)}}=a$
Now, write the logarithm given in question.
$\Rightarrow {{\log }_{x}}25$ = -0.5
Form the exponents:
$\Rightarrow {{x}^{{{\log }_{x}}\left( 25 \right)}}={{x}^{-0.5}}$
By applying ${{x}^{{{\log }_{x}}\left( a \right)}}=a$, we will get:
$\Rightarrow 25={{x}^{-0.5}}$
As we know the multiplicative inverse of ${{a}^{-x}}=\dfrac{1}{{{a}^{x}}}$, so apply this and solve further:
$\Rightarrow 25=\dfrac{1}{\sqrt{x}}$ (0.5 is $\dfrac{1}{2}$ which means the under root)
Keep the under root x alone to find the value of x:
$\Rightarrow \sqrt{x}=\dfrac{1}{25}$
Now, square the both sides to remove the under root:
$\Rightarrow {{\left( \sqrt{x} \right)}^{2}}={{\left( \dfrac{1}{25} \right)}^{2}}$
Solve further by removing brackets:
$\therefore x=\dfrac{1}{625}$
This is the final answer.

Note: You have to deal with the exponents very carefully. You should know the square and square of any number. For squares, a number has to be multiplied 2 times and for square root, pairing is done. Before solving any question, go through all the formulae of logarithm because according to them only you can form the equation.