Question

If In a = 2, ln b = 3, and ln c = 5, evaluate the following:

(a) In
-2
a
b¹c-2
(b) In √b²c¹a-3
(c)
= 3
=
In(a²6³)
In (bc)-³
(d) (In c¹) (in)
502
||
>
>
OF
8
OF

Answer 1

`(a) \(\ln\left((1)/(e^3)\right) = -3\),`

(b)
`\(\ln\left(\sqrt{e^(8)}\right) = 8\),`

(c)
`\(\ln\left(-(1)/(54)\right) = -\ln(54)\),`

(d)
`\((\ln 502) = \ln 502\).`

Let's break down and evaluate each expression using the given values:

Given:

`\[ a = 2, \quad \ln b = 3, \quad \ln c = 5 \]`

(a)
`\[ \ln\left((a^(-2))/(b^(1)c^(-2))\right) \]`

Substitute the values:

`\[ \ln\left((2^(-2))/(e^3 \cdot e^(-2))\right) \]`

Simplify:

`\[ \ln\left((1)/(e^(3-2+2))\right) \]`

`\[ \ln\left((1)/(e^3)\right) \]`

`\[ \ln(e^(-3)) \]`

`\[ -3 \]`

`(b) \[ \ln\left(\sqrt{b^2c^1a^(-3)}\right) \]`

Substitute the values:

`\[ \ln\left(\sqrt{e^6 \cdot e^5 \cdot e^(-3)}\right) \]`

Simplify:

`\[ \ln\left(\sqrt{e^(6+5-3)}\right) \]`

`\[ \ln(e^8) \]`

`\[ 8 \]`

`\[ \ln\left((a^2)/(6^3)\right) \]`

Substitute the values:

`\[ \ln\left((2^2)/(6^3)\right) \]`

Simplify:

`\[ \ln\left((4)/(216)\right) \]`

`\[ \ln\left((1)/(54)\right) \]`

`\[ \ln(54^(-1)) \]`

`\[ -\ln(54) \]`

`\[ (\ln c^1) \cdot (\ln 502) \]`

Substitute the values:

`\[ (5^1) \cdot (\ln 502) \]`

`\[ \ln 502 \]`

In summary:

(a)
`\(-3\)`

(b)
`\(8\)`

(c)
`\(-\ln(54)\)`

(d)
`\(\ln 502\)`