$\log_{3}{(5x-2)}$ $-$ $2\log_{3}{\sqrt{3x+1}}$ $\,=\,$ $1-\log_{3}{4}$ is a logarithmic equation. It is developed in mathematics by taking number $3$ as base of the logarithms.
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.
$\implies$ $\log_{3}{(5x-2)}$ $-$ $\log_{3}{(3x+1)}$ $+$ $\log_{3}{4}$ $\,=\,$ $1$
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.
$\implies$ $\log_{3}{\Bigg(\dfrac{4(5x-2)}{3x+1}\Bigg)}$ $\,=\,$ $1$
Write the logarithmic equation in exponential form equation by the mathematical relation between logarithms and exponents.
$\implies$ $\dfrac{4(5x-2)}{3x+1}$ $\,=\,$ $3^1$
$\implies$ $\dfrac{4(5x-2)}{3x+1}$ $\,=\,$ $3$
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.
A best free mathematics education website for students, teachers and researchers.
Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.
Learn how to solve the math problems in different methods with understandable steps and worksheets on every concept for your practice.
Copyright © 2012 - 2022 Math Doubts, All Rights Reserved