solve logarithm of x to the base 10 + logarithm of X - 15 to the base 10 equal to 2
Answers
Answer:
Use the log property that log(a)+log(b)=log(ab), so
log(x−15)+log(x)=2
log(x(x−15))=2
Since no base was specified we assume the base of the logarithm was 10, take the base-10 exponential of both sides, i.e.:
10log(x(x−15))=102
We know that blogb(a)=a, and that 102=100 so
(x(x−15))=100
Expand the product and take that 100 to the LHS
x2−15x−100=0
Solve the quadratic in your favorite way, by the quadratic formula for example
x=15+_under root of 15^2-4×1×(-100)
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2×1
=15+_under root of 225+400
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2
=15+_under root625
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2
=15+25/2 = 15-25/2
=40/2 = (-10/2)
=20 =(-5)
However, recall that this was originally a logarithmic formula; we can have anything equal to 0 or a negative number in one!
So we know that
x-15>0
x>0
20-15=5>0 and 20>0so it's a valid solution, but −5<0 so it isn't one. So the set of solutions is S={20}