Question

Lg(x-3)+lg(2x+1)-lg(5x-3)=0

Answer 1

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Answer 2

To solve the equation lg(x - 3) + lg(2x + 1) - lg(5x - 3) = 0, we can use logarithmic properties to simplify and solve for x.

First, recall the properties of logarithms:

1. lg(a) + lg(b) = lg(ab)
2. lg(a) - lg(b) = lg(a/b)

Using these properties, we can simplify the equation:

lg(x - 3) + lg(2x + 1) - lg(5x - 3) = 0

Combine the first two logarithms using property 1:

lg((x - 3)(2x + 1)) - lg(5x - 3) = 0

Now, we have a single logarithm on the left side of the equation. To eliminate the logarithm, we can rewrite the equation in exponential form:

(x - 3)(2x + 1) = 5x - 3

Now, let's solve for x:

Expand and simplify the left side of the equation:

2x^2 - 5x - 3 = 5x - 3

Now, move all terms to one side of the equation:

2x^2 - 5x - 3 - 5x + 3 = 0

Combine like terms:

2x^2 - 10x = 0

Factor out 2x:

2x(x - 5) = 0

Now, set each factor equal to zero and solve for x:

1. 2x = 0 x = 0

2. x - 5 = 0 x = 5

So, there are two possible solutions for the equation:

x = 0 and x = 5

You can check both solutions by substituting them back into the original equation to make sure they satisfy it.