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Logarithmic Equations Calculator

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1

Solved example of logarithmic equations

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

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

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

Rearrange the equation

$\log \left(x-1\right)+\log \left(1000\right)=\log \left(x+1\right)$
4

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

$\log \left(1000\left(x-1\right)\right)=\log \left(x+1\right)$
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 $b$

$1000\left(x-1\right)=x+1$
6

Move everything to the left hand side of the equation

$1000\left(x-1\right)-x-1=0$
7

Multiply the single term $1000$ by each term of the polynomial $\left(x-1\right)$

$1000x-1000-x-1=0$
8

Subtract the values $-1000$ and $-1$

$-1001+1000x-x=0$
9

Combining like terms $1000x$ and $-x$

$-1001+999x=0$
10

We need to isolate the dependent variable $x$, we can do that by subtracting $-1001$ from both sides of the equation

$999x=1001$
11

Divide both sides of the equation by $999$

$x=1.002002$

Verify that the solutions obtained are valid in the initial equation

12

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

$x=1.002002$

Final Answer

$x=1.002002$

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