Question #cae1a

2 Answers
NickTheTurtle
Apr 12, 2017

#x=5#

Explanation:

Use the property #log(a)+log(b)=log(ab)# to simplify the equation to #log((x+1)*(x-1))=log(24)#.

This implies that #(x+1)*(x-1)=24#. Then, #x^2-1=24#. Add #1# to both sides: #x^2=25#. There are two solutions, #x=+-5#.

However, if we substitute #-5# back into the equation, we would be taking the logarithm of a negative number, which is not allowed. We can eliminate #-5# as an answer.

The only real solution is #5#.

Alan P.
Apr 12, 2017

#color(green)(x=5)#

Explanation:

Remember:
[1]#color(white)("XXX")log(a)+log(b)=log(a * b)#
[2]#color(white)("XXX")log(c)# is only defined for #c > 0#

#log(x+1)+log(x-1)#
#color(white)("XXX")=log(x^2-1) =log(24)#

#rarrx^2-1=24#

#rarr x^2=25#

#rarr x=+-5#

...but neither #log(x+1)# nor #log(x-1)# are defined if #x=-5#,
so this is an extraneous solution,

leaving only
#color(white)("XXX")x=5#

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