Solve $\log_{3}{\big(5+4\log_{3}{(x-1)}\big)}$ $\,=\,$ $2$

In this logarithmic equation problem, a logarithmic expression is defined in terms of $x$ on the left-hand side of the equation and it is given that the value of the logarithmic expression is equal to $2$.

$\log_{3}{\big(5+4\log_{3}{(x-1)}\big)}$ $\,=\,$ $2$

Now, let us learn how to solve the logarithmic equation to find the value of $x$.

Convert log equation into exponential form

It is possible to solve the given logarithmic equation only when the algebraic equation is released from the logarithm. Hence, use the mathematical relation between the exponents and logarithms to transform the given logarithmic equation into exponential form.

$\implies$ $5+4\log_{3}{(x-1)}$ $\,=\,$ $3^2$

Simplify the algebraic equation

Now, it is time to simplify the algebraic equation. The equation can be simplified by using the fundamental operations.

$\implies$ $5+4\log_{3}{(x-1)}$ $\,=\,$ $9$

$\implies$ $4\log_{3}{(x-1)}$ $\,=\,$ $9-5$

$\implies$ $4\log_{3}{(x-1)}$ $\,=\,$ $4$

$\implies$ $\log_{3}{(x-1)}$ $\,=\,$ $\dfrac{4}{4}$

$\implies$ $\log_{3}{(x-1)}$ $\,=\,$ $\dfrac{\cancel{4}}{\cancel{4}}$

$\implies$ $\log_{3}{(x-1)}$ $\,=\,$ $1$

Solve the logarithmic equation

The simplified equation is once again in logarithmic form. So, use the mathematical relationship between the exponents and logarithms one more time to release the logarithmic equation from logarithm.

$\implies$ $\log_{3}{(x-1)}$ $\,=\,$ $1$

$\implies$ $x-1$ $\,=\,$ $3^1$

Now, solve the algebraic equation to find the value of $x$.

$\implies$ $x-1$ $\,=\,$ $3$

$\implies$ $x$ $\,=\,$ $3+1$

$\,\,\,\therefore\,\,\,\,\,\,$ $x$ $\,=\,$ $4$

Verification

$\log_{3}{\big(5+4\log_{3}{(x-1)}\big)}$ $\,=\,$ $2$

It is solved that the value of $x$ is equal to $4$. Now, let’s verify the value of $x$ by substituting in the logarithmic expression.

$=\,\,\,$ $\log_{3}{\big(5+4\log_{3}{(4-1)}\big)}$

$=\,\,\,$ $\log_{3}{\big(5+4\log_{3}{(3)}\big)}$

The logarithm of $3$ to base $3$ is equal to $1$ as per the fundamental rule of the logarithms.

$=\,\,\,$ $\log_{3}{(5+4(1))}$

$=\,\,\,$ $\log_{3}{(5+4 \times 1)}$

$=\,\,\,$ $\log_{3}{(5+4)}$

$=\,\,\,$ $\log_{3}{(9)}$

The base in the logarithm is $3$. So, the number $9$ can be expressed in exponential form in terms of $3$.

$=\,\,\,$ $\log_{3}{\big(3^2\big)}$

Now, use the power law of the logarithms to find the value of the expression.

$=\,\,\,$ $2 \times \log_{3}{3}$

$=\,\,\,$ $2 \times 1$

$=\,\,\,$ $2$

Therefore, the value $x$ equals to $4$ is the real solution of the given logarithmic equation.