Question

How does one solve ${5.6^{3m}} = 20$ ?

Answer 1

Hint: In the given question, we have been asked to find the value of ‘m’ and it is given that ${5.6^{3m}} = 20$. To solve this question, we need to get ‘m’ on one side of the “equals” sign, and all the other numbers on the other side. To solve this equation for a given variable ‘m’, we have to undo the mathematical operations such as addition, subtraction, multiplication, and division that have been done to the variables.

Formula used: We used logarithm property i.e.,
$\log {a^b} = b\log a$ .

Complete step by step solution:
It is given that ${5.6^{3m}} = 20$ ,
We have to solve for $m$ ,
Taking ${5.6^{3m}} = 20$,
Divide both the side of the equation by $5$ ,
We get ,
$
   \Rightarrow \dfrac{{{{5.6}^{3m}}}}{5} = \dfrac{{20}}{5} \\
   \Rightarrow {6^{3m}} = 4 \\
 $
Now taking logarithm of base $10$ both the side ,
$ \Rightarrow \log {}_{10}({6^{3m}}) = {\log _{10}}(4)$
Using the logarithm property that is $\log {a^b} = b\log a$ ,
We will get ,
$ \Rightarrow 3m\log {}_{10}(6) = {\log _{10}}(4)$
Divide both the side of the equation by ${\log _{10}}(6)$ ,
We will get ,
$ \Rightarrow \dfrac{{3m\log {}_{10}(6)}}{{\log {}_{10}(6)}} = \dfrac{{{{\log }_{10}}(4)}}{{{{\log }_{10}}(6)}}$
$ \Rightarrow 3m = \dfrac{{{{\log }_{10}}(4)}}{{{{\log }_{10}}(6)}}$
$ \Rightarrow 3m = 0.7737$
Now , divide both the side of the equation by $3$,
We will get ,
$
   \Rightarrow \dfrac{{3m}}{3} = \dfrac{{0.7737}}{3} \\
   \Rightarrow m = 0.2579 \\
 $
Therefore, the value of $m$ is equal to $0.2579$ .
It is the required answer.

Additional information: In the given question , mathematical operations such as addition, subtraction, multiplication and division is used. Use addition or subtraction properties of equality to gather variable terms on one side of the equation and constant on the other side of the equation. Use the multiplication or division properties of equality to form the coefficient of the variable term equivalent to one.

Note: The important thing to recollect about any equation is that the ‘equals’ sign represents a balance. What the sign says is that what’s on the left-hand side is strictly an equal to what’s on the right-hand side. It is the type of question where only mathematical operations such as addition, subtraction, multiplication and division is used.