Question

If ln =2, ln =3, and ln =5, evaluate the following:

Answer 1

Using Basic Property of Logarithms:

(a) ln ( a⁻² / b c⁴ ) = -27

(b) ln $\sqrt{b^4 c^4 a^4}$ = 20

(c) ln ( a³ b⁻²) / ln (bc)³ = 0

(d) ( ln c⁻⁴ ) x ( ln (a/b⁻³) ) = -220

Basic property of Logarithms:

1. Product Property

log(mn) = log(m) + log(n)

2. Quotient Property

log(m/n) = log(m) – log(n)

3. Power rule

logmⁿ = n (log(m))

some property for Natural Log (ln).

here, ln a = 2, ln b = 3 and ln c = 5

(a) ln ( a⁻² / b c⁴ )

using Quotient property:

ln a⁻² - ln ( b c⁴)

now using product and power rule property:

-2( ln a) - ln(b) - 4 ( ln c)

putting the values

-2x(2) - 3 - 4 (5)

-4 - 3 - 20

→ -27

(b) ln $\sqrt{b^4 c^4 a^4}$

using power rule property:

1/2 ( ln ( b⁴c⁴a⁴ ))

using Product property and Power rule property:

4/2 ( ln b + ln c + ln a)

2 ( 3 + 5 + 2 )

2 ( 10 )

→ 20

(c) ln ( a³ b⁻²) / ln (bc)³

using Product and Power rule property:

3( ln a) -2(ln b) / 3 ( ln b + ln c )

3 (2) -2(3) / 3 ( 3 + 5)

→ 0

(d) ( ln c⁻⁴ ) x ( ln (a/b⁻³) )

using Power rule and Quotient property:

-4 ( ln c) x ( ln a + 3 ( ln b ))

-4 ( 5 ) x ( 2 + 3 ( 3 ) )

(-20) x ( 11 )

→ -220

Hence, Using Basic Property of Logarithms:

(a) ln ( a⁻² / b c⁴ ) = -27

(b) ln $\sqrt{b^4 c^4 a^4}$ = 20

(c) ln ( a³ b⁻²) / ln (bc)³ = 0

(d) ( ln c⁻⁴ ) x ( ln (a/b⁻³) ) = -220

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