Question

How do you solve $10\log x = 15$ ?

Answer 1

Hint : The above question is based on the concept of logarithms. The main approach towards solving the expression is by applying the logarithm properties. On applying the properties of logarithm and further on simplifying we will get a value for the above expression.

Complete step-by-step answer :
Logarithm is the exponent or power to which base must be raised to yield a given number. When expressed mathematically x is the logarithm of base n to the base b if ${b^x} = n$ ,then we can write it has
$x = {\log _b}n$
The simplest way to understand the logarithm is given for example-Number of 2s we multiply to get 8.
$2 \times 2 \times 2 = 8$ . So, we have to multiply 3 of the 2s to get 8. Therefore, the number of 2s we need to
multiply to get 8 is 3.
${\log _2}\left( 8 \right) = 3$
So now in the above given expression
$\log x = \dfrac{{15}}{{10}}$
Further reducing it into lowest terms we get,
$\log x = \dfrac{3}{2}$
Now if $y = {10^x}$ then $\log \left( y \right) = x$ .Since $y = {10^x}$ .Therefore we can write it has $\log \left( {{{10}^x}} \right) = x$
Now by taking exponents we get
$x = {10^{\log x}}$
Since we already have the value of logx then after substituting it.
$x = {10^{\log x}} = {10^{\dfrac{3}{2}}}$
Now the fraction which is obtained in the power should be written in such a way that it can further be reduced.
So,it can be written as
$1 + \dfrac{1}{2} = \dfrac{3}{2}$
Therefore,
${10^{1 + \dfrac{1}{2}}} = 10\sqrt {10}$
Therefore, we get the above value.
So, the correct answer is “$10\sqrt {10}$”.

Note : An important thing to note is that whenever there is an addition of terms in the power we can always split the base according to the terms in the power.For example the the fraction $1 + \dfrac{1}{2} = \dfrac{3}{2}$
can be written and split the base 10 as ${10^1}$ and ${10^{\dfrac{1}{2}}}$ .