Question

1. multiple choice \\( \ln x^5 = \\)\

(a) \\( 5 \ln x \\)\
(b) \\( 2 \ln x^3 \\)\
(c) \\( x \ln 5 \\)\
(d) \\( 3 \ln x^2 \\)\
(e) \\( \ln x^2 \cdot \ln x^3 \\)

Explanation:

Step1: Apply logarithm power rule

The logarithm power rule is \(\ln a^b = b\ln a\). For \(\ln x^5\), here \(a = x\) and \(b = 5\).
So, \(\ln x^5=5\ln x\).

Step2: Check other options

• Option B: \(2\ln x^3 = 2\times3\ln x=6\ln x

eq5\ln x\).

• Option C: \(x\ln5\) is not related to \(\ln x^5\) by logarithm rules.
• Option D: \(3\ln x^2=3\times2\ln x = 6\ln x

eq5\ln x\).

• Option E: \(\ln x^2\cdot\ln x^3=(2\ln x)\cdot(3\ln x)=6(\ln x)^2

eq5\ln x\).

Answer:

A. \(5\ln x\)