Question

The value of $1-2\log 5+3\log 2$ is $\log \dfrac{16}{x}$. Find $x$.

Answer 1

Hint: We will use the logarithmic formulas like, $a\log b=\log {{b}^{a}}$, $\log a+\log b=\log (ab)$, $\log a-\log b=\log \dfrac{a}{b}$, etc to find the value of $x$. Also, we will consider $1$ as $\log 10$ because the value of $\log 10$ is $1$.
Complete step-by-step answer:
It is given in the question that using the expression $1-2\log 5+3\log 2$, we have to find the value of $x$ in $\log \dfrac{16}{x}$.
$\Rightarrow 1-2\log 5+3\log 2$ …………………………………………….$(1)$
To evaluate this expression we will use logarithmic formulas, as we know that
$\begin{align}
& a\log b=log{{b}^{a}} \\
& \log a+\log b=\log (ab) \\
& \log a-\log b=\log \dfrac{a}{b} \\
\end{align}$
Using the formula, $a\log b=\log {{b}^{a}}$ we can write $2\log 5$ as $\log {{(5)}^{2}}$. Similarly, we can write $3\log 2$ as $\log {{(2)}^{3}}$ in equation $(1)$, we get
$\Rightarrow 1-\log {{(5)}^{2}}+\log {{(2)}^{3}}$
$\Rightarrow 1-\log (25)+\log (8)$ ………………………………………….$(2)$
As we discussed that we can write $1$ as $\log 10$. So, on replacing $1$ with $\log 10$ in equation $(2)$, we get
$\Rightarrow \log 10-\log 25+\log 8$
$\Rightarrow \log 10+\log 8-\log 25$ ………………………………………$(3)$
Now, we will use formula $\log a+\log b=\log (ab)$ in equation $(3)$, we get
$\Rightarrow \log (10\times 8)-\log 25$ …………………………………………$(4)$
Now, we will use the formula $\log a-\log b=\log \dfrac{a}{b}$ in equation $(4)$, we get
$\Rightarrow \log \dfrac{(10\times 8)}{25}$
$\Rightarrow \log \dfrac{80}{25}$ …………………………………………………………….$(5)$
Dividing numerator and denominator with $5$ in equation $(5)$, we get
$\Rightarrow \log \dfrac{16}{5}$ ……………………………………………………………..$(6)$
$\Rightarrow \log (3.2)$
And we know that the value of $\log (3.2)$ is $0.50514$. But, in question we have to find $x$ in $\log \dfrac{16}{x}$
Comparing $\log \dfrac{16}{x}$ with equation $(6)$, we get
$\Rightarrow \log \dfrac{16}{x}=\log \dfrac{16}{5}$.
Therefore, $x=5$

Note: using the logarithmic formulas wherever required will reduce your steps and efforts to solve the question. Try to memorize all the common logarithmic formulas. $a\log b=\log {{b}^{a}}$, $\log a+\log b=\log (ab)$, $\log a-\log b=\log \dfrac{a}{b}$ etc. So that we can solve the question in a few steps as shown below.
$\begin{align}
& 1+2\log 5-3\log 2=\log \dfrac{(10\times 8)}{25} \\
& \Rightarrow \log \dfrac{16}{5}. \\
\end{align}$