Math Doubts

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

$\log_{3}{(5x-2)}$ $-$ $2\log_{3}{\sqrt{3x+1}}$ $\,=\,$ $1-\log_{3}{4}$ is a logarithmic equation. It is developed in mathematics by taking number $3$ as base of the logarithms.

Eliminate square root from term

The square root of $3x+1$ can be eliminated from second term by the multiply factor $2$ as exponent of the $3x+1$. It can be done by using power rule of logarithms.

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

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

Make the logarithmic equation to have log terms one side and constant term in other side of the equation.

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

Combine Logarithmic terms

A negative sign between first two log terms represents subtraction of log terms. They can be combined by using quotient rule of logarithms.

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

A plus sign between the log terms expresses a summation of them. They can be merged as a logarithmic term by the product rule of logarithms.

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

Transform Equation in Exponential form

Write the logarithmic equation in exponential form equation by the mathematical relation between logarithms and exponents.

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

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

Solve the equation

Use cross multiplication method to solve the algebraic equation and it evaluates the value of $x$.

$\implies$ $4(5x-2)$ $\,=\,$ $3(3x+1)$

$\implies$ $4 \times 5x-4 \times 2$ $\,=\,$ $3 \times 3x + 3 \times 1$

$\implies$ $20x-8$ $\,=\,$ $9x+3$

$\implies$ $20x-9x$ $\,=\,$ $3+8$

$\implies$ $11x$ $\,=\,$ $11$

$\implies$ $x$ $\,=\,$ $\dfrac{11}{11}$

$\implies$ $x$ $\,=\,$ $\require{cancel} \dfrac{\cancel{11}}{\cancel{11}}$

$\,\,\, \therefore \,\,\,\,\,\, x$ $\,=\,$ $1$

Thus, the log equation $\log_{3}{(5x-2)}$ $-$ $2\log_{3}{\sqrt{3x+1}}$ $\,=\,$ $1-\log_{3}{4}$ is solved by properties of logarithms in logarithmic mathematics.

Math Doubts
Math Doubts is a free math tutor for helping students to learn mathematics online from basics to advanced scientific level for teachers to improve their teaching skill and for researchers to share their research projects. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more