I would like to express $\dfrac{\ln\left(\frac{1}{2}\right)}{80000000}$ as a negative exponent. Using Desmos online calculator I tried this variant:
Why are the results different? I expected/goal was to be the same result.
How can I express $\dfrac{\ln\left(\frac{1}{2}\right)}{80000000}$ not as a fraction?
$\endgroup$ 22 Answers
$\begingroup$\begin{align} \frac{\ln(\frac12)}{8\cdot 10^7} &= \frac{1}{8\cdot 10^7}\cdot \ln(\frac12)\\ &= \frac{1}8{\cdot 10^{-7}}\cdot \ln(\frac12)\\ &=\ln\left((\frac12)^{\frac{1}8{\cdot 10^{-7}}}\right)\\ &\ne\ln\left((\frac12)^{-8{\cdot 10^{7}}}\right) \end{align}
$\endgroup$ $\begingroup$$$a\ln b=\ln\left(b^a\right)$$
Using this fact, we can write
$$\frac{\ln\left(\frac12\right)}{80000000}=\frac1{80000000}\ln\left(\frac12\right)=\ln\left(\left(\frac12\right)^{\frac1{80000000}}\right)=\ln\left(\frac{1^{\frac1{80000000}}}{2^{\frac1{80000000}}}\right)=\ln\left(2^{-\frac1{80000000}}\right)$$
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