Question

Solve the following equation:
${\log _3}\left( {1 + {{\log }_3}\left( {{2^x} - 7} \right)} \right) = 1$

Answer 1

Here we are given an equation and we need to solve it. We can note that the given equation contains logarithmic terms. So, we need to remove the logarithmic terms so that we are able to find the solutions for the given equations. To remove the logarithmic terms, we need to convert log into antilog.

Complete step by step answer:
The given equation is ${\log _3}\left( {1 + {{\log }_3}\left( {{2^x} - 7} \right)} \right) = 1$ and we are asked to solve it.
Before getting into the solution, we just learn how to convert log to antilog.
The first step is to note the base of the given logarithm. The next step is to raise both sides to that base and this step removes the logarithm.
For example, let us consider $y = {\log _{10}}\left( 3 \right)$ .
Now, we need to note the base of the logarithm.
Here the base of the logarithm is $10$. Then, we shall raise both sides to the base$10$.
Thus, we get ${10^y} = 3$
Now, we shall get into our solution.
The given equation is ${\log _3}\left( {1 + {{\log }_3}\left( {{2^x} - 7} \right)} \right) = 1$.
Now, we need to remove the log. Here the base is $3$ and we need to raise both sides to the base $3$.
Thus, we get $1 + {\log _3}\left( {{2^x} - 7} \right) = {3^1}$
$\Rightarrow {\log _3}\left( {{2^x} - 7} \right) = 3 - 1$
$\Rightarrow {\log _3}\left( {{2^x} - 7} \right) = 2$
Similarly, we need to remove this logarithm too.
We note that the base of the log is $3$ and we need to raise both sides to the base $3$.
Thus, we get ${2^x} - 7 = {3^2}$
$\Rightarrow {2^x} - 7 = 9$
$\Rightarrow {2^x} = 9 + 7$
$\Rightarrow {2^x} = 16$
$\Rightarrow {2^x} = {2^4}$ (Here$16 = {2^4}$)
The base is equal on both sides in the above equation; hence we can compare the powers.
Therefore, we have $x = 4$.

Note: Since the given equation contains logarithmic terms, we need to convert them into antilog. First, we need to note the base of the logarithmic term and then we need to raise the base on both sides. Hence, we have removed the logarithm. And, we got the solution $x = 4$ for the given equation.