Question

How do you solve ${{2}^{x+4}}=\dfrac{1}{32}$?

Answer 1

Hint: In this question we will first convert the right-hand side into a term which has a power of $2$ and then use the property of similar bases of logarithm to simplify the equation and then solve for the value of $x$.

Complete step by step answer:
We have the given equation as ${{2}^{x+4}}=\dfrac{1}{32}$
now the term in the right-hand side can be written in the form of exponent as:
$\Rightarrow {{2}^{x+4}}=\dfrac{1}{{{2}^{5}}}$
now we know the property of exponents that $\dfrac{1}{{{a}^{n}}}={{a}^{-n}}$therefore, on using this property of the right-hand side of the expression, we get:
$\Rightarrow {{2}^{x+4}}={{2}^{-5}}$
On taking the logarithm on both the sides, we get:
 $\Rightarrow {{\log }_{10}}{{2}^{x+4}}={{\log }_{10}}{{2}^{-5}}$
Now we know the property of logarithm that ${{\log }_{10}}{{a}^{b}}=b{{\log }_{10}}a$, we can write the equation as:
$\Rightarrow (x+4){{\log }_{10}}2=(-5){{\log }_{10}}2$
Now on dividing both the sides by ${{\log }_{10}}2$, we get:
\[\Rightarrow \dfrac{(x+4){{\log }_{10}}2}{{{\log }_{10}}2}=\dfrac{(-5){{\log }_{10}}2}{{{\log }_{10}}2}\]
On simplifying the expression, we get:
$\Rightarrow x+4=-5$
On transferring the term $4$from the left-hand side to the right-hand side, we get:
$\Rightarrow x=-5-4$
On simplifying, we get:
$\Rightarrow x=-9$, which is the required solution.

Note: Now to cross check whether the answer of $x$ is correct, we will put the value of$x$in the left-hand side of the expression to get the value, on substituting, we get:
 $\Rightarrow {{2}^{-9-4}}$
On simplifying the exponent term, we get:
$\Rightarrow {{2}^{-5}}$
This term can be written in the reciprocal form as:
$\Rightarrow \dfrac{1}{{{2}^{5}}}$
Which can be simplified as:
$\Rightarrow \dfrac{1}{32}$, which is the right-hand side, therefore the solution is correct.
It is to be noted that the logarithm we are using has the base $10$, the base is the number to which the log value has to be raised to, to get the original term. This is also called the antilog of the number which is the logical reverse of taking a log.
The most commonly used bases in logarithm are $10$ and $e$ which has a value of approximate $2.713...$
Logarithm is used to simplify a mathematical expression, it converts multiplication to addition, division to subtraction and exponents to multiplication.