Here, we show you a step-by-step solved example of logarithmic equations. This solution was automatically generated by our smart calculator:
Express the numbers in the equation as logarithms of base $10$
Any expression to the power of $1$ is equal to that same expression
The sum of two logarithms of the same base is equal to the logarithm of the product of the arguments
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 $b$
Multiplying polynomials $x$ and $x-3$
When multiplying two powers that have the same base ($x$), you can add the exponents
Multiplying polynomials $x$ and $x-3$
Move everything to the left hand side of the equation
Factor the trinomial $x^2-3x-10$ finding two numbers that multiply to form $-10$ and added form $-3$
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 $-5$ from both sides of the equation
Canceling terms on both sides
Combining all solutions, the $2$ solutions of the equation are
Verify that the solutions obtained are valid in the initial equation
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
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