Quadratic Logarithmic equations

Definition

A logarithmic equation that transforms as a quadratic equation while solving, is called a quadratic logarithmic equation.

Introduction

The equations are formed sometimes in logarithmic form and it is very common to solve them in mathematics. In some cases, The logarithmic equations are surprisingly transformed as quadratic equations while solving them because the quadratic form equations are internally hidden in logarithmic equations and this type of logarithmic equation is called the logarithmic equation in quadratic form. It is simply called the quadratic logarithmic equation.

$(1).\,\,$ $\log{(x+1)}$ $+$ $\log{(x-1)}$ $\,=\,$ $\log{8}$

$(2).\,\,$ $\log_{3}{2}$ $+$ $2\log_{3}{x}$ $\,=\,$ $\log_{3}{(7x-3)}$

$(3).\,\,$ $\log_{5}{x}$ $+$ $\log_{5}{(x-3)}$ $\,=\,$ $\log_{5}{10}$

The above three examples express the equations in logarithmic form. We can easily identify them as the logarithmic equations but we cannot identify that the quadratic equations are internally hidden in them. It will be revealed while solving the log equation.

Solution

There are three simple steps to solve the logarithmic equations in the form of a quadratic equation.

1. Simplify the logarithmic equation to maximum level by logarithmic identities.
2. Remove the logarithmic form from the equation when you identify the quadratic equation.
3. Finally, solve the quadratic equation to find the variable in the logarithmic equation.

Example

Now, let’s learn what a quadratic form logarithmic equation is in mathematics from an understandable simple example, and learn how it can be solved by the concept of quadratic equations.

Solve $\log_{3}{2}$ $+$ $2\log_{3}{x}$ $\,=\,$ $\log_{3}{(7x-3)}$

Firstly, simplify the logarithmic equation by the logarithmic formulas.

$\implies$ $\log_{3}{2}$ $+$ $\log_{3}{x^2}$ $\,=\,$ $\log_{3}{(7x-3)}$

$\implies$ $\log_{3}{(2 \times x^2)}$ $\,=\,$ $\log_{3}{(7x-3)}$

$\implies$ $\log_{3}{(2x^2)}$ $\,=\,$ $\log_{3}{(7x-3)}$

Now, remove the logarithmic form from the equation by equating equals quantities on both sides of the equation.

$\implies$ $2x^2$ $\,=\,$ $7x-3$

Finally, solve the quadratic equation to evaluate the variable.

$\implies$ $2x^2$ $-7x$ $+$ $3$ $\,=\,$ $0$

$\implies$ $x$ $\,=\,$ $\dfrac{-(-7)\pm \sqrt{(-7)^2-4 \times 2 \times 3}}{2 \times 2}$

$\implies$ $x$ $\,=\,$ $\dfrac{7\pm \sqrt{49-24}}{4}$

$\implies$ $x$ $\,=\,$ $\dfrac{7\pm \sqrt{25}}{4}$

$\implies$ $x$ $\,=\,$ $\dfrac{7 \pm 5}{4}$

$\implies$ $x$ $\,=\,$ $\dfrac{7+5}{4}$ or $x$ $\,=\,$ $\dfrac{7-5}{4}$

$\implies$ $x$ $\,=\,$ $\dfrac{12}{4}$ or $x$ $\,=\,$ $\dfrac{2}{4}$

$\implies$ $x$ $\,=\,$ $\dfrac{\cancel{12}}{\cancel{4}}$ or $x$ $\,=\,$ $\dfrac{\cancel{2}}{\cancel{4}}$

$\,\,\,\therefore\,\,\,\,\,\,$ $x$ $\,=\,$ $3$ or $x$ $\,=\,$ $\dfrac{1}{2}$

Therefore, the solution set for the giving logarithmic equation is $\bigg\{\dfrac{1}{2},\, 3\bigg\}$

Worksheet

$(1).\,\,$ Solve $\log_{5-x}{(x^2-2x+65)} \,=\, 2$

$(2).\,\,$ Solve $2\log_{\large x}{a}$ $+$ $\log_{\large ax}{a}$ $+$ $3\log_{\large a^2x}{a}$ $\,=\,$ $0$

$(3).\,\,$ Solve $\log_{10}{\big(98+\sqrt{x^2-12x+36}\big)}$ $\,=\,$ $2$

The list of questions on solving the quadratic logarithmic equations with answers for your practice and solutions to learn how to solve the quadratic form logarithmic equations by the log rules.

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