Question

log(3x²+7)=log(3x-1)=1

Answer 1

⟹
log
7
+
log
(
3
x
−
2
)
=
log
(
x
+
3
)
+
log
10
Now, use product law of logarithm and combine both left hand side and right hand sides of the equations as single logarithmic terms.

⟹
log
(
7
×
(
3
x
−
2
)
)
=
log
(
(
x
+
3
)
×
10
)
⟹
log
(
7
(
3
x
−
2
)
)
=
log
(
10
(
x
+
3
)
)
This mathematical statement clears that logarithm of an expression is equal to logarithmic of another expression. Therefore, they two algebraic equations should be equal mathematically.

⟹
7
(
3
x
−
2
)
=
10
(
x
+
3
)
Now, simplify this algebraic equation to obtain the value of the
x
.

⟹
21
x
−
14
=
10
x
+
30
⟹
21
x
−
10
x
=
14
+
30
⟹
11
x
=
44
⟹
x
=
44
11
⟹
x
=
44
11
∴
x
=
4
Therefore, the value of the
x
is
4
and the required solution for this logarithm problem.

Verification
Substitute
x
=
4
in left hand side of the equation.

log
7
+
log
(
3
x
−
2
)
=
log
7
+
log
(
3
(
4
)
−
2
)
⟹
log
7
+
log
(
3
x
−
2
)
=
log
7
+
log
(
12
−
2
)
⟹
log
7
+
log
(
3
x
−
2
)
=
log
7
+
log
(
10
)
⟹
log
7
+
log
(
3
x
−
2
)
=
log
(
7
×
10
)
∴
log
7
+
log
(
3
x
−
2
)
=
log
70
Now, substitute the value of
x
in right hand side of the equation.

log
(
x
+
3
)
+
1
=
log
(
4
+
3
)
+
1
⟹
log
(
x
+
3
)
+
1
=
log
7
+
log
10
⟹
log
(
x
+
3
)
+
1
=
log
(
7
×
10
)
∴
log
(
x
+
3
)
+
1
=
log
70
It is proved that, for
x
=
4