Question

if 3^x=m ,then the value of 10^x log base 10 ^3+2 log base 10 ^3 is​

Answer 1

Property of Logarithms

ⓐ $\log_{a}p+\log_{a}q=\log_{a}pq\ \text{[Addition of Logarithms]}$

ⓑ $\log_{a}p-\log_{a}q=\log_{a}\dfrac{p}{q}\ \text{[Subtraction of Logarithms]}$

ⓒ $\log_{a}p^{q}=q\log p\ \text{[Logarithms of Exponents]}$

ⓓ $\log_{a}p=\dfrac{\log_{t}p}{\log_{t}a}\ \text{[Change of Base Formula]}$

Appropriate Question

If $3^{x}=m$, then the value of $10^{x\log_{10}3+2\log_{10}3}$.

$\text{(Given\ Condition)}$

① $3^{x}=m\Leftrightarrow x=\log_{3}m$

Solution

$\text{(Given Expression)}$

$=10^{x\log_{10}3+2\log_{10}3}$

Let's apply the property of logarithms.

$\log_{10}\text{(Given Expression)}$

$=\log_{10}10^{x\log_{10}3+2\log_{10}3}$

$=\boxed{(x\log_{10}3+2\log_{10}3)}\log_{10}10$ [by ⓒ]

$=\boxed{\log _{3}m}\cdot \log_{10}3+2\log_{10}3$ [by ①]

$=\boxed{\dfrac{\log_{10} m}{\cancel{\log_{10}3}} \cdot \dfrac{\cancel{\log_{10}3}}{\log_{10}10} }+2\log_{10}3$ [by ⓓ]

$=\dfrac{\log_{10}m}{\log_{10}10}+2\log_{10}3$

$=\boxed{\log_{10}m}+2\log_{10}3$ [by ⓓ]

$=\log_{10}m+\log_{10}\boxed{3^{2}}$ [by ⓒ]

$=\log_{10}\boxed{9m}$ [by ⓐ]

$\log_{10}\text{(Given Expression)}=\log_{10}9m$

$\implies \text{(Given Expression)}=9m$

So, the required answer is $9m$.