π Part I β Solving Equations
We use log β exponential conversion and log properties to solve. There are two main equation types:
Single log/exp
Isolate β convert form β solve
Isolate β convert form β solve
Multiple logs
Condense to one side β Form 1 or Form 2
Condense to one side β Form 1 or Form 2
π Steps for Single Log or Exponential
1
Isolate the log or exponential expression on one side
2
Convert to the alternate form (log β exponential)
3
Solve for the variable
4
Simplify and leave in exact form (unless told to round)
βοΈ Example 1 β Single Exponential
a)
4Λ£ β 2 = 7
4Λ£ = 9
4Λ£ = 9
x = logβ 9
b)
(2)x+3 + 5 = 26
(2)x+3 = 21
(2)x+3 = 21
x = logβ 21 β 3
c)
βex = 3 β 4ex
3ex = 3 β ex = 52
3ex = 3 β ex = 52
x = ln(52)
βοΈ Example 2 β Single Log
a)
2 logβ(x+3) + 4 = 10
logβ(x+3) = 3
x+3 = 27
logβ(x+3) = 3
x+3 = 27
x = 24
b)
2 + logβ
(2xβ1) = 4
logβ (2xβ1) = 2
2xβ1 = 25
logβ (2xβ1) = 2
2xβ1 = 25
x = 13
c)
ln(2βx)/5 = 1
ln(2βx) = 5
2βx = eβ΅
ln(2βx) = 5
2βx = eβ΅
x = 2 β eβ΅
π Equations with Multiple Logarithms
Two End Forms
Form 1: logb(f) = c
β Convert to exponential: f = bc
β Convert to exponential: f = bc
Form 2: logb(f) = logb(g)
β Set equal: f = g
β Set equal: f = g
Always check for extraneous solutions! After solving, verify that all arguments of logarithms in the original equation are positive. Discard any solutions that make a log argument β€ 0.
βοΈ Example 3 β Multiple Logs (Form 2)
a)
log(3)+log(x+4)=log(5xβ2)
log[3(x+4)] = log(5xβ2)
3x+12 = 5xβ2
log[3(x+4)] = log(5xβ2)
3x+12 = 5xβ2
x = 7 β
b)
ln(x+1)βln(3xβ5)=ln7
ln[(x+1)/(3xβ5)] = ln7
(x+1)/(3xβ5) = 7
ln[(x+1)/(3xβ5)] = ln7
(x+1)/(3xβ5) = 7
x = 9/5 β
c)
logβ(x+2)+logβ(xβ3)=logβ14
(x+2)(xβ3)=14
xΒ²βxβ20=0 β x=5 or x=β4
(x+2)(xβ3)=14
xΒ²βxβ20=0 β x=5 or x=β4
x = 5 only (β4 fails domain)
d)
ln(x)βln(3)=2
ln(x/3)=2
x/3 = eΒ²
ln(x/3)=2
x/3 = eΒ²
x = 3eΒ²
βοΈ Example 4 β Check for No Solution
a)
logβ(x)β3logβ2=logβ(x+7)
logβ(x/8)=logβ(x+7)
x/8 = x+7 β x = β8
logβ(x/8)=logβ(x+7)
x/8 = x+7 β x = β8
No solution (x must be > 0)
b)
log(x)+3log(2)=log(2xβ7)
log(8x)=log(2xβ7)
8x = 2xβ7 β x = β76
log(8x)=log(2xβ7)
8x = 2xβ7 β x = β76
No solution (x must be > 72)
βοΈ Examples 5β6 β Intersection Points (Quadratic)
Ex 5
ln(x+10)βln(x)=ln(x+3)
ln[(x+10)/x]=ln(x+3)
x+10 = x(x+3) = xΒ²+3x
xΒ²+2xβ10=0
ln[(x+10)/x]=ln(x+3)
x+10 = x(x+3) = xΒ²+3x
xΒ²+2xβ10=0
x = (β2+β44)/2 β 2.317
(negative root rejected)
Ex 6
logβ(2xβ3)+logβ(x+4)=logβ(8x+2)
(2xβ3)(x+4)=8x+2
2xΒ²β3xβ14=0
(2xβ7)(x+2)=0
(2xβ3)(x+4)=8x+2
2xΒ²β3xβ14=0
(2xβ7)(x+2)=0
x = 72 only (β2 fails domain)
βοΈ Example 7 β Common Base (Multiple Exponentials)
a)
25xβ1 = 22x+7
5xβ1 = 2x+7
x = 8/3
b)
83xβ2 = 2x+5
(2Β³)3xβ2 = 2x+5
9xβ6 = x+5
(2Β³)3xβ2 = 2x+5
9xβ6 = x+5
x = 11/8
c)
92x = 27x+2
(3Β²)2x = (3Β³)x+2
4x = 3x+6
(3Β²)2x = (3Β³)x+2
4x = 3x+6
x = 6
d)
(1/2)5x+7 = 4x
2β5xβ7 = 22x
β5xβ7 = 2x
2β5xβ7 = 22x
β5xβ7 = 2x
x = β1
π Part II β Finding Inverses of Exp & Log Functions
Steps to Find f β»ΒΉ(x)
1
Switch x and y in the equation
2
Use inverse operations to solve for y (reverse order of operations)
3
Write the final answer as f β»ΒΉ(x) = ...
Ex 1: Find f β»ΒΉ
f(x) = 3(2)x+4 β 15
y = 3(2)x+4 β 15
β switch xβy:
x = 3(2)y+4 β 15
(x+15)/3 = 2y+4
logβ((x+15)/3) = y+4
y = 3(2)x+4 β 15
β switch xβy:
x = 3(2)y+4 β 15
(x+15)/3 = 2y+4
logβ((x+15)/3) = y+4
fβ»ΒΉ(x) = logβ((x+15)/3) β 4
Ex 2: Find g β»ΒΉ
g(x) = β3logβ
(xβ4) + 6
β switch xβy:
x = β3logβ (yβ4)+6
(6βx)/3 = logβ (yβ4)
5(6βx)/3 = yβ4
β switch xβy:
x = β3logβ (yβ4)+6
(6βx)/3 = logβ (yβ4)
5(6βx)/3 = yβ4
gβ»ΒΉ(x) = 5(6βx)/3 + 4
π Part II β Solving Inequalities
Strategy: Isolate the expression, use log β exp conversion, then apply a sign chart at critical x-values. Check domain restrictions first!
Example 4 β Log Inequality
f(x) = log((xΒ²β4x+7)/(x+3)) > 0 [Domain: x > β3]
log((xΒ²β4x+7)/(x+3)) > 0
β argument must be > 1 (since log > 0 means argument > 10β° = 1)
xΒ²β4x+7 > x+3
xΒ²β5x+4 > 0
(xβ1)(xβ4) > 0
log((xΒ²β4x+7)/(x+3)) > 0
β argument must be > 1 (since log > 0 means argument > 10β° = 1)
xΒ²β4x+7 > x+3
xΒ²β5x+4 > 0
(xβ1)(xβ4) > 0
Solution: β3 < x < 1 or x > 4
Example 5 β Compare Two Log Functions
g(x) = ln(xΒ²+xβ5) < h(x) = ln(x+4)
ln((xΒ²+xβ5)/(x+4)) < 0 β argument < 1
xΒ²+xβ5 < x+4 β xΒ²β9 < 0 β (x+3)(xβ3) < 0
ln((xΒ²+xβ5)/(x+4)) < 0 β argument < 1
xΒ²+xβ5 < x+4 β xΒ²β9 < 0 β (x+3)(xβ3) < 0
Solution: β3 < x < (β1+β21)/2 or (β1+β21)/2 < x < 3 (with domain constraints)
Example 6 β Exponential Inequality
k(x) = 1 β 4(2)xβ3 β€ β31
β4(2)xβ3 β€ β32
(2)xβ3 β₯ 8 = 2Β³
xβ3 β₯ 3
β4(2)xβ3 β€ β32
(2)xβ3 β₯ 8 = 2Β³
xβ3 β₯ 3
Solution: x β₯ 6
π Extended β FRQ #4 Style (No Calculator)
β οΈ AP Exam FRQ #4 β No Calculator Allowed
These problems require exact answers obtained through algebraic manipulation. Common approach: isolate, use log/exp properties, simplify carefully.
FRQ 1
f(x) = ex/4 / e3/5
Set f(x) = e7/4:
(x4) β (35) = 74
x/4 = 74 + 35 = 43/20
Set f(x) = e7/4:
(x4) β (35) = 74
x/4 = 74 + 35 = 43/20
x = 43/5
FRQ 2
g(x) = 8e3x β ex
Set g(x) = 3ex:
8e3x = 4ex
e3x/ex = 1/2
e2x = 1/2
2x = ln(1/2) = βln2
Set g(x) = 3ex:
8e3x = 4ex
e3x/ex = 1/2
e2x = 1/2
2x = ln(1/2) = βln2
x = βln2/2 = (1βln2)/3
Careful: rewrite step-by-step
FRQ Tip: Leave answers in exact form (e.g., ln(52), not a decimal). Rewrite using properties before combining. Show all steps β partial credit is awarded for correct work even if the final answer has an error.
Strategy
What are the steps to solve an equation with a single log or exponential?
Tap to reveal β¨
Answer
Isolate β Convert form (logβexp) β Solve β Simplify
Step 1: get the log or exp alone. Step 2: rewrite in the other form. Step 3: solve for x. Step 4: leave exact form unless told to round.
Topic 2.13
Exponential & Log Equations
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Question 1 of 5
Solve for x (exact form): 4Λ£ β 2 = 7
Isolate the exponential, then convert to log form
Question 2 of 5
Solve: 2 logβ(x + 3) + 4 = 10
Isolate the log, then convert to exponential form
Question 3 of 5
Solve: logβ(x+2) + logβ(xβ3) = logβ14. Check for extraneous solutions.
Product Property, then set equal, then factor
Question 4 of 5
Solve: 92x = 27x+2 using common bases.
Rewrite 9 = 3Β² and 27 = 3Β³, then set exponents equal
Question 5 of 5
Find f β»ΒΉ(x) for f(x) = 3(2)x+4 β 15.
Switch x and y, isolate the exponential, convert to log
0/5
Keep going!