Logarithmic Equations Calculator

Solved example of logarithmic equations

We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $-\log \left(x+6\right)$ from both sides of the equation

Canceling terms on both sides

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

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 $b$

Move everything to the left hand side of the equation

Factor the trinomial $x^2-x-6$ finding two numbers that multiply to form $-6$ and added form $-1$

Thus

Break the equation in $2$ factors and set each equal to zero, to obtain

Solve the equation ($1$)

We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $2$ from both sides of the equation

Canceling terms on both sides

Solve the equation ($2$)

We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $-3$ from both sides of the equation

Canceling terms on both sides

Combining all solutions, the $2$ solutions of the equation are

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