Step-by-step Solution

Solve the logarithmic equation $2\log \left(x\right)-\log \left(x+6\right)=0$

Go!
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Final Answer

$x=3$

Step-by-step solution

Problem to solve:

$2\cdot log\left(x\right)-1\cdot log\left(x+6\right)=0$
1

Apply the formula: $a\log_{b}\left(x\right)$$=\log_{b}\left(x^a\right)$, where $a=2$ and $b=10$

$\log \left(x^2\right)-\log \left(x+6\right)=0$
2

The difference of two logarithms of equal base $b$ is equal to the logarithm of the quotient: $\log_b(x)-\log_b(y)=\log_b\left(\frac{x}{y}\right)$

$\log \left(\frac{x^2}{x+6}\right)=0$
3

Rewrite the number $0$ as a logarithm of base $10$

$\log \left(\frac{x^2}{x+6}\right)=\log \left(1\right)$
4

For two logarithms of the same base to be equal, their arguments must be equal. In other words, if $\log(a)=\log(b)$ then $a$ must equal $b$

$\frac{x^2}{x+6}=1$
5

Multiply both sides of the equation by $x+6$

$x^2=x+6$
6

Grouping terms

$x^2-x=6$
7

Rewrite the equation

$x^2-x-6=0$
8

To find the roots of a polynomial of the form $ax^2+bx+c$ we use the quadratic formula, where in this case $a=1$, $b=-1$ and $c=-6$. Then substitute the values of the coefficients of the equation in the quadratic formula:

  • $\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$x=\frac{1\pm 5}{2}$
9

To obtain the two solutions, divide the equation in two equations, one when $\pm$ is positive ($+$), and another when $\pm$ is negative ($-$)

$x=\frac{1+5}{2},\:x=\frac{1-5}{2}$
10

Subtract the values $1$ and $-5$

$x=\frac{1+5}{2},\:x=\frac{-4}{2}$
11

Add the values $1$ and $5$

$x=\frac{6}{2},\:x=\frac{-4}{2}$
12

Divide $6$ by $2$

$x=3,\:x=\frac{-4}{2}$
13

Divide $-4$ by $2$

$x=3,\:x=-2$

Verify that the solutions obtained are valid in the initial equation

14

The valid solutions to the logarithmic equation are the ones that, when replaced in the original equation, don't result in any logarithm of negative numbers or zero, since in those cases the logarithm does not exist

$x=3$

Final Answer

$x=3$
$2\cdot log\left(x\right)-1\cdot log\left(x+6\right)=0$

Main topic:

Logarithmic Equations

Related Formulas:

1. See formulas

Time to solve it:

~ 0.07 s