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# Logarithmic Equations Calculator

## Get detailed solutions to your math problems with our Logarithmic Equations step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here.

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###  Difficult Problems

1

Here, we show you a step-by-step solved example of logarithmic equations. This solution was automatically generated by our smart calculator:

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

Express the numbers in the equation as logarithms of base $10$

$\log \left(x\right)+\log \left(x-3\right)=\log \left(10^{1}\right)$
3

Any expression to the power of $1$ is equal to that same expression

$\log \left(x\right)+\log \left(x-3\right)=\log \left(10\right)$
4

The sum of two logarithms of the same base is equal to the logarithm of the product of the arguments

$\log \left(x\left(x-3\right)\right)=\log \left(10\right)$
5

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 $b$

$x\left(x-3\right)=10$

Multiplying polynomials $x$ and $x-3$

$x\cdot x-3x=10$

When multiplying two powers that have the same base ($x$), you can add the exponents

$x^2-3x=10$
6

Multiplying polynomials $x$ and $x-3$

$x^2-3x=10$
7

Move everything to the left hand side of the equation

$x^2-3x-10=0$
8

Factor the trinomial $x^2-3x-10$ finding two numbers that multiply to form $-10$ and added form $-3$

$\begin{matrix}\left(2\right)\left(-5\right)=-10\\ \left(2\right)+\left(-5\right)=-3\end{matrix}$
9

Thus

$\left(x+2\right)\left(x-5\right)=0$
10

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

$x+2=0,\:x-5=0$
11

Solve the equation ($1$)

$x+2=0$
12

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

$x+2-2=0-2$
13

Canceling terms on both sides

$x=-2$
14

Solve the equation ($2$)

$x-5=0$
15

We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $-5$ from both sides of the equation

$x-5+5=0+5$
16

Canceling terms on both sides

$x=5$
17

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

$x=-2,\:x=5$

Verify that the solutions obtained are valid in the initial equation

18

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

The equation has no solutions.

##  Final answer to the problem

The equation has no solutions.

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