Question

If ${\log _8}m = 3.5$ and ${\log _2}n = 7$ , then find the value of m in terms of n.

Answer 1

Hint:
First we will use the following definition of logarithmic function that is the expression ${\log _a}b = n$ signifies the following expression that ${a^n} = b$ , now we will try to convert each of the logarithmic expression into the later equation and solve for m and n.

Complete step by step solution:
Given ${\log _8}m = 3.5$ and ${\log _2}n = 7$
If \[{\log _b}a = x\] , then we have \[a = {b^x}\]
Using this we get,
 $ \Rightarrow {\log _8}m = 3.5$
The above expression signifies that ${8^{3.5}} = m$ …(i)
Similarly, for ${\log _2}n = 7$ which implies that
 $ \Rightarrow {2^7} = n$ …(ii)
The expression (i) can also be written as
 $ \Rightarrow {2^{7 \times \left( {\dfrac{{3.5 \times 3}}{7}} \right)}} = m$
Which can also be written as
 $ \Rightarrow {\left( {{2^7}} \right)^{\left( {\dfrac{3}{2}} \right)}} = m$ …(iii)
Using the equation (ii) in equation (iii) we get
 $ \Rightarrow m = {n^{\left( {\dfrac{3}{2}} \right)}}$

Hence, we got our relation in which the value m is written in terms of n.

Note:
We can also use the properties of logarithms to solve the above problem ,but in doing so we should check the value of the variables we will use in the logarithmic function , for example the variable used as a in ${\log _b}a$ should not be less than or equal to zero , because function is not defined in that case.