HOW TO EVALUATE LOGARITHMS WITH FRACTIONS

To evaluate logarithmic function, we should know how to convert the logarithmic function into exponential form.

Evaluate each of the following logarithms without the use of a calculator.

Problem 1 :

log₄ 1/2 =

Solution :

log₄ 1/2 = x

4^x = 1/2

(2²)^x = 1/2

(2)^2x = 2^-1

2x = -1

x = -1/2

Problem 2 :

log₆ 1/36 =

Solution :

log₆ 1/36 = x

6^x = 1/36

6^x = 1/6²

6^x = 6^-2

x = -2

Problem 3 :

log_2/3 9/4 =

Solution :

log_2/3 9/4 = x

(2/3)^x = 9/4

(2/3)^x = (3/2)²

(2/3)^x = (2/3)^-²

x = -2

Problem 4 :

log₈₁ 1/27 =

Solution :

log₈₁ 1/27 = x

81^x = 1/27

(3⁴)^x = 1/3³

(3)^4x = 3^-3

4x = -3

x = -3/4

Problem 5 :

log₄ 1/8 =

Solution :

log₄ 1/8 = x

4^x = 1/8

(2²)^x = 1/2³

(2)^2x = 2^-3

2x = -3

x = -3/2

Problem 6 :

log₃₆ 1/6 =

Solution :

log₃₆ 1/6 = x

36^x = 1/6

(6²)^x = 1/6

(6)^2x = 6^-1

2x = -1

x = -1/2

Problem 7 :

log_1/16 32 =

Solution :

log_1/16 32 = x

(1/16)^x = 32

(1/2⁴)^x = 2⁵

(2)^-4x = 2⁵

-4x = 5

x = -5/4

Problem 8 :

log₂₇ 1/3 =

Solution :

log₂₇ 1/3 = x

27^x = 1/3

(3³)^x = 1/3

(3)^3x = 3^-1

3x = -1

x = -1/3

Problem 9 :

log_1/4 16 =

Solution :

log_1/4 16 = x

(1/4)^x = 16

(1/2²)^x = 2⁴

(2)^-2x = 2⁴

-2x = 4

x = -4/2

x = -2

Problem 10 :

Expand the logarithmic expression :

ln (5/12x)

Solution :

= ln (5/12x)

= ln 5 - ln 12x

= ln 5 - [ln 12 + ln x]

= ln 5 - ln 12 - ln x

Problem 11 :

Use the properties of logarithms to condense each expression :

a) 3 log 4 − 2 log 𝑘

b) −5 log₂ (𝑥 +1) + 3 log₂ (6𝑥)

c) (1/3) log₄ 10 + (1/3) log₄ ℎ − 6 log₄ 𝑔

d) ln (3𝑚 +5) − 4 ln 𝑚 − ln (𝑚 − 1)

e) log 20 + 2 log (1/2) − log 𝑥 + 3 log 𝑦

Solution :

a) 3 log 4 − 2 log 𝑘

= log 4³  − log k²

= log 64  − log k²

= log (64/k²)

b) −5 log₂ (𝑥 +1) + 3 log₂ (6𝑥)

= 3 log₂ (6𝑥) − 5 log₂ (𝑥 + 1)

= log₂ (6𝑥)³ − log₂ (𝑥 + 1)⁵

= log₂ (6)³ (𝑥)³ − log₂ (𝑥 + 1)⁵

= log₂ (216𝑥³) − log₂ (𝑥 + 1)⁵

= log₂ [(216𝑥³) / (𝑥 + 1)⁵]

c) (1/3) log₄ 10 + (1/3) log₄ ℎ − 6 log₄ 𝑔

= log₄ ∛10 + log₄ ∛ℎ − log₄ 𝑔⁶ 

= log₄ (∛10 ∛ℎ) − log₄ 𝑔⁶ 

= log₄ (∛10ℎ) − log₄ 𝑔⁶ 

= log₄ [∛(10ℎ) / 𝑔⁶]

d) ln (3𝑚 + 5) − 4 ln 𝑚 − ln (𝑚 − 1)

= ln (3m + 5) − ln 𝑚⁴ − ln (𝑚 − 1)

= ln (3m + 5) − [ln 𝑚⁴ + ln (𝑚 − 1)]

= ln (3m + 5) − [ln 𝑚⁴ (𝑚 − 1)]

= ln [(3m + 5) / 𝑚⁴ (𝑚 − 1)]

e) log 20 + 2 log (1/2) − log 𝑥 + 3 log 𝑦

= log 20 + log (1/2)² − log 𝑥 + log 𝑦³

= log 20 + log (1/4) − log 𝑥 + log 𝑦³

= log 20 ⋅ (1/4) ⋅ 𝑦³ − log 𝑥

= log 5 𝑦³ − log 𝑥

= log (5 𝑦³/𝑥)

Problem 12 :

Use properties of logarithms to expand each expression. The expanded logarithm expressions should have arguments with no exponent, product, or quotient

a) ln (4/5) 

b) log₆3x

c) log 7b/√c

d) log₂(m³/8n)

e) log₃[(u - 1)/(v⁵w³)]

Solution :

a) ln (4/5) = ln 4 - ln 5

b) log₆3x = log₆3 + log₆x

c) log 7b/√c = log 7b - log √c

= log 7 + log b - log c^1/2

= log 7 + log b - (1/2) log c

d) log₂(m³/8n) = log₂m³-  log₂8n

= log₂m³-  [log₂8 + log₂ n]

= 3log₂m - [log₂2³+ log₂ n]

= 3log₂m - [3log₂2+ log₂ n]

= 3log₂m - [3(1) + log₂ n]

= 3log₂m - 3 - log₂ n

e) log₃[(u - 1)/(v⁵w³)]

= log₃(u - 1) - log₃(v⁵w³)

= log₃(u - 1) - [log₃v⁵ + log₃w³]

= log₃(u - 1) - [5 log₃v + 3 log₃w]

= log₃(u - 1) - 5 log₃v - 3 log₃w

Problem 13 :

Solve each equation.

log₄ x = log₄ 3 + log₄ (x − 2)

Solution :

log₄ x - log₄ (x − 2) = log₄ 3

log₄ (x/(x − 2)) = log₄ 3

x/(x - 2) = 3

x = 3(x - 2)

x = 3x - 6

x - 3x = -6

-2x = -6

x = 6/2

x = 3

So, the value of x is 3.

Problem 14 :

ln(𝑥 − 3) + ln(2𝑥 + 3) = ln (−4𝑥²)

Solution :

ln(𝑥 − 3) + ln(2𝑥 + 3) = ln (−4𝑥²)

ln(x - 3)(2x + 3) = ln (−4𝑥²)

(x - 3)(2x + 3) = −4𝑥²

2x² + 3x - 6x - 9 = −4𝑥²

2x² + 4𝑥² - 3x - 9 = 0

6x² - 3x - 9 = 0

2x² - x - 3 = 0

a = 2, b = -1 and c = -3

x = [-b ± √(b²- 4ac)]/2a

x = [1 ± √(1²- 4(2)(-3))]/2(2)

= [1±√(1 + 24)]/4

= [1±√25]/4

= [1 ± 5]/4

x = (1 + 5)/4 and x = (1 - 5)/4

x = 6/4 and x = -4/4

x = 3/2 and x = -1

Problem 15 :

log (3𝑥 + 2) = 1 + 𝑙𝑜𝑔 2𝑥

Solution :

log (3𝑥 + 2) = 1 + 𝑙𝑜𝑔 2𝑥

log (3𝑥 + 2) - 𝑙𝑜𝑔 2𝑥 = 1

log [(3x + 2)/2x] = 1

(3x + 2)/2x = 10¹

(3x + 2)/2x = 10

3x + 2 = 10(2x)

3x + 2 = 20x

3x - 20x = -2

-17x = -2

x = 2/17