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 Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved