SME
Grade 11 Math · Problem type
Solve a log equation
Use this for “Solve log₃(2x − 1) = 4.”
Chart
Vertical asymptote u = 0
f(u) = g(u)
The curves never cross, so the equation has no solution.
This is an example. Type your own values to start a new problem.
log
(
a
x
+
(
c
)
)
=
y
a
x
+
(
c
)
=
b
=
u
x
Solution
worked out (not typed)
Formulas
u
=
b
y
81
=
3
4
x
=
u
−
c
a
41
=
81 − (−1)
2
Assumptions
•
Rewrite logb(u) = y as u = bʸ, with u = ax + c: the log is an exponent.
•
Then solve ax + c = bʸ for x.
•
Check: ax + c must be positive, and it is, since bʸ is.
Step-by-step
WE KNOW
b = 3, a = 2, c = −1, y = 4
FIND
inside of the log, ax + c (u), solution (x)
Step 1 · Find inside of the log, ax + c (u)
Use u = bʸ
logb(
u
)
=
y
means
bʸ
=
u
.
u
=
3
4
u
=
81
Step 2 · Find solution (x)
Use x = (u − c) ÷ a
Solve
ax
+
c
=
u
:
take
away
c
,
then
divide
by
a
.
x
=
81 − (−1)
2
x
=
82
2
x
=
41
Check
Put the numbers back in. Both sides should match.
81
=
3
4
✓
41
=
81 − (−1)
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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Main lesson
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Use this for “$2,000 grows at 5% a year compounded continuously. How long until it reaches $3,000?”
Equations with two logs
Use this for “Solve log₃ x + log₃(x − 8) = 2.”