Solve each of the following equations. Write all answers in simplest form.

Question

log₆x+log₆(x+1)=1

 

log₆(x+2)+log₆(x+3)=1

Answer 1

Using the properties of logs, we can condense the equations so that we have a single log.  Also, the solution to a logarithm is the exponent of the log's base number.

 

1)

 

log₆(x(x + 1)) = 1

 

log₆(x² + x) = 1

 

6¹ = x² + x

 

6 = x² + x

 

 

Subtract 6 on both sides of the equation.

 

0 = x² + x - 6

 

 

Now you have a quadratic equation.   Factoring, we get

 

0 = (x + 3)(x - 2)

 

 

Solving for x from this equation.

 

x = -3           and          x = 2

 

 

Plug in these values into the equation to validate them.

 

 

2)

 

log₆((x + 2)(x + 3)) = 1

 

log₆(x² + 5x + 6) = 1

 

6¹ = x² + 5x + 6

 

6 = x² + 5x + 6

 

x² + 5x = 0

 

 

 

We have a quadratic equation again.  Factoring, we get

 

x(x + 5) = 0

 

 

Solve for x.  I leave this part to you to finish.