SOLVING LOGARITHMIC EQUATIONS WORKSHEET

Problem 1 : 

Solve  for x :

log₅(x) = 1/2

Problem 2 : 

Solve for y :

log_1/5(x) = 3

Problem 3 :

Solve for z :

log_z8√2 = 7

Problem 4 : 

Solve for k :

-3 = log_k0.001

Problem 5 :

Solve for x :

log₅[5log₃(x)] = 2

Problem 6 :

Solve for m :

m + 2log₂₇(9) = 0

Problem 7 :

If log₂(x) + log₄(x) + log₁₆(x) = 21/4, then what is the value of x?

Problem 8 :

If 2log(y) = 4log(3),  then find the value of y. 

Problem 9 :

If 3p is equal to log(0.3) to the base 9, then find the value of p.

Problem 10 :

Solve for r :

Problem 11 :

Solve for q :

log₃(q) + log₉(q) + log₈₁(q) = 7/4

Problem 12 :

Solve for v :

log₄(v + 4) + log₄8 = 2

Problem 13 :

Solve for w :

log₆(w + 4) - log₆(w - 1) = 1

Problem 14 :

Solve for g :

log₂(g) + log₄(g) + log₈(g) = 11/6

Problem 15 :

Solve for x :

81^√x = log₂(512)

Answers

1. Answer :

log₅(x) = 1/2

Convert the equation to exponential form.

x = 5^1/2

x = √5

2. Answer :

log_1/5(y) = 3

Convert the equation to exponential form.

y = (1/5)³

y = 1³/5³

y = 1/125

3. Answer :

log_z8√2 = 7

Convert the equation to exponential form.

8√2 = z⁷

 2 ⋅ 2 ⋅ 2 ⋅ √2 = x⁷

Each 2 can be expressed as (√2 ⋅ √2).

Then,

√2 ⋅ √2 ⋅ √2 ⋅ √2 ⋅ √2 ⋅ √2 ⋅ √2 = x⁷

√2⁷ = x⁷

Because the exponents are equal, bases can be equated. 

x = √2

4. Answer :

-3 = log_k0.001

Convert the equation to exponential form.

k^-3 = 0.001

1/k³ = 1/1000

Take reciprocal on both sides. 

k³ = 1000 

k³ = 10³

Because the exponents are equal, bases can be equated. 

k = 10

5. Answer :

log₅[5log₃(x)] = 2

Convert it to exponential form. 

5log₃(x) = 5²

5log₃(x) = 25

Divide both sides by 5.

log₃(x) = 5

Convert the equation to exponential form.

x = 3⁵

x = 243

6. Answer :

m + 2log₂₇(9) = 0

m = -2log₂₇(9)

m = log₂₇(9^-2)

Convert the equation to exponential form.

27^m = 9^-2

(3³)^m = (3²)^-2

3^3m = 3^-4

Because the bases are equal, exponents can be equated. 

3m = -4

m = -4/3

7. Answer :

8. Answer :

2log(y) = 4log(3)

Divide each side by 2.

log(y) = 2log(3)

log(y) = log(3²)

log(y) = log(9)

y = 9

9. Answer :

From the information given, we have

3p = log₉(0.3)

Solve for p.

3p = log₉(1/3)

3p = log₉(1) - log₉(3)

3p = 0 - log₉(3)

3p = -log₉(3)

3p = -½log₃(3)

3p = -½(1)

3p = -½

Divide both sides by 3.

p = -1/6

10. Answer :

7r - 4 = 5(r + 2)

7r - 4 = 5r + 10

2r - 4 = 10

2r = 14

r = 7

11. Answer :

log₃(q) + log₉(q) + log₈₁(q) = 7/4

log₃(q) + ½log₃(q) + ¼log₃(q) = 7/4

log₃(q) ⋅ (1 + ½ + ¼) = 7/4

log₃(q) ⋅ 7/4 = 7/4

Multiply both sides by 4/7.

log₃(q) = 1

Convert the equation to exponential form.

q = 3¹

q = 3

12. Answer :

log₄(v + 4) + log₄8 = 2

Use the Product Rule.

log₄[8(v + 4)] = 2

log₄(8v + 32) = 2

Convert the equation to exponential form.

8v + 32 = 4²

8v + 32 = 16

8v = -16

v = -2

13. Answer :

log₆(w + 4) - log₆(w - 1) = 1

Use the Quotient Rule.

log₆[(w + 4)/(w - 1)] = 1

Convert the equation to exponential form.

(w + 4)/(w - 1) = 6¹

(w + 4)/(w - 1) = 6

w + 4 = 6(w - 1)

w + 4 = 6w - 6

-5w + 4 = -6

-5w = -10

w = 2

14. Answer :

log₂(g) + log₄(g) + log₈(g) = 11/6

log₂(g) + ½log₂(g) + ⅓log₂(g) = 11/6

log₂(g) ⋅ (1 + ½ + ⅓) = 11/6

log₂(g) ⋅ 11/6 = 11/6

Multiply both sides by 6/11.

log₂(g) = 1

Convert the equation to exponential form.

g = 2¹

g = 2

15. Answer :

81^√x = log₂(512)

81^√x = log₂(2⁹)

81^√x = 9log₂(2)

(9²)^√x = 9(1)

9^2√x = 9¹

2√x = 1

Divide both sides by 2.

√x = 1/2

Square both sides.

(√x)² = (1/2)²

x = 1/4

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