Question

Solve log x-1=-log (x-9)

Answer 1

log
x
+
log
(
x
−
9
)
=
1

Recall that
log
a
+
log
b
=
log
(
a
b
)
, so
log
x
+
log
(
x
−
9
)
=
log
(
x
(
x
−
9
)
)
=
log
(
x
2
−
9
x
)

log
(
x
2
−
9
x
)
=
1

Convert the logarithmic equation to an exponential equation.
10
log
(
x
2
−
9
x
)
=
10
1

Remember that
10
log
x
=
x
, so
x
2
−
9
x
=
10

x
2
−
9
x
−
10
=
0

(
x
−
10
)
(
x
+
1
)
=
0

x
−
10
=
0
and
x
+
1
=
0

x
=
10
and
x
=
−
1

Check:
log
x
+
log
(
x
−
9
)
=
1

If
x
=
10
,
log
10
+
log
(
10
−
9
)
=
1

log
10
+
log
1
=
1

1
+
0
=
1

1
=
1

x
=
10
is a solution.
If
x
=
−
1
,
log
(
−
1
)
+
log
(
−
1
−
9
)
=
1

This is impossible, because the logarithm of a negative number is undefined.

Answer 2

Answer:

At the first of the question it is given that -log(x-9) which is impossible because the negative log of a number is undefined