Copyright

Why is \frac{\ln(0.5)}{\ln(2)} = -1, (without using a calculator)?

Question:

Why is {eq}\frac{\ln(0.5)}{\ln(2)} = -1, {/eq} (without using a calculator)?

Answer and Explanation:

The given logarithm expression is:

{eq}\displaystyle \frac{\ln(0.5)}{\ln(2)} = -1 {/eq}

On simplifying the left hand side of the above expression, we get:

{eq}\begin{align*} \displaystyle \frac{\ln(0.5)}{\ln(2)} &= \frac{\ln(\frac{5}{10})}{\ln(2)}\\ &= \frac{\ln(\frac{5}{5\times 2})}{\ln(2)}\\ &= \frac{\ln(\frac{1}{2})}{\ln(2)}\\ &= \frac{\ln(1)-\ln(2)}{\ln(2)} &\textrm{Use Quotient rule}\\ &= \frac{0-\ln(2)}{\ln(2)} &\because \ln(1)=0\\ &= -\frac{\ln(2)}{\ln(2)}\\ &=-1\\ &=\textrm{Right hand side}\\ \end{align*} {/eq}

L.H.S=R.H.S

Hence proved.


Learn more about this topic:

Logarithmic Function: Definition & Examples

from Precalculus: Homework Help Resource

Chapter 2 / Lesson 10
49K

Recommended Lessons and Courses for You

Explore our homework questions and answer library