# 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

Solved example of logarithmic equations

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

Grouping terms

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

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

$\log \left(\frac{x+1}{x-1}\right)=3$
4

Rewrite the number $3$ as a logarithm of base $10$

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

$\frac{x+1}{x-1}=1000$
6

Multiply both sides of the equation by $x-1$

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

Solve the product $1000\left(x-1\right)$

$x+1=1000x-1000$
8

We need to isolate the dependent variable $x$, we can do that by subtracting $1$ from both sides of the equation

$x=-1001+1000x$
9

Grouping terms

$-999x=-1001$
10

Divide both sides of the equation by $-999$

$x=1.002002$

Verify that the solutions obtained are valid in the initial equation

11

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

$x=1.002002$

$x=1.002002$