Question

How do you solve the equation $\log x = \log 5 + 3\log 3$?

Answer 1

This problem deals with solving logarithms. Here some basic properties and identities of logarithms are used for simplifying the given equation which is in terms of logarithmic expressions, and linear in the variable $x$. The formulas or the properties of logarithms which are used to solve the given problem are:
$\Rightarrow \log ab = \log a + \log b$
$\Rightarrow n\log a = \log {a^n}$
$\Rightarrow {a^{{{\log }_a}b}} = b$

Complete step-by-step solution:
Consider the given logarithmic equation below as shown:
$\Rightarrow \log x = \log 5 + 3\log 3$
Now consider the second term of the above equation which is $3\log 3$ as shown below:
$\Rightarrow 3\log 3$
Applying the basic property of the logarithms which is $n\log a = \log {a^n}$ to the above expression as shown below:
$\Rightarrow 3\log 3 = \log {3^3}$
Now substituting this simplified above expression in the given logarithmic equation as shown below:
$\Rightarrow \log x = \log 5 + 3\log 3$
$\Rightarrow \log x = \log 5 + \log {3^3}$
We know that the value of ${3^3} = 27$, so substituting this in the above expression as shown below:
$\Rightarrow \log x = \log 5 + \log 27$
Now consider the right hand side of the above equation which is $\log 5 + \log 27$, now applying the basic property of logarithms which is $\log a + \log b = \log \left( {ab} \right)$, as shown below:
$\Rightarrow \log x = \log 5 + \log 27$
$\Rightarrow \log x = \log 5\left( {27} \right)$
Now simplifying the above expression as shown below:
$\Rightarrow {\log _e}x = {\log _e}\left( {135} \right)$
So on taking the exponents on both sides of $e$, as shown below:
$\Rightarrow {e^{{{\log }_e}x}} = {e^{{{\log }_e}\left( {135} \right)}}$
We know the basic property of logarithms which are ${a^{{{\log }_a}x}} = x$, applying this property to the above expression as shown below:
$\Rightarrow {e^{{{\log }_e}x}} = {e^{{{\log }_e}\left( {135} \right)}}$
$\therefore x = 135$
The solution of $x$ in $\log x = \log 5 + 3\log 3$ is equal to 135.

Note: Please note that the above problem is solved with the help of applications of some basic identities and properties of logarithms, such as the multiplication rule of logarithms, and the exponential rule of the logarithms. There are few more important properties of logarithms such as division rule and few, such as:
$\Rightarrow \log \left( {\dfrac{a}{b}} \right) = \log a - \log b$
The basic logarithmic property is, if ${\log _a}x = b$, then the value of $x = {a^b}$.