0
$\begingroup$

I'm following a probability calculation that results in $20^{150}$ and this result is converted into base $10$ which answers $10^{195}$. I don't understand how they obtained that answer? Can someone please help explain how to convert? Thanks.

|cite|improve this question
$\endgroup$
  • $\begingroup$ Your title does not match the body. I assume the body is correct. The title sounds like you want a $1$ and $150$ zeros. $\endgroup$ – Ross Millikan Sep 2 '18 at 2:42
  • $\begingroup$ $150 \log_{10} 2 \approx 45.15445, $ so $2^{150} \approx 1.427 \cdot 10^{45}$ $\endgroup$ – Will Jagy Sep 2 '18 at 2:46
2
$\begingroup$

$$20^{150}=e^{150 \ln 20}=e^{150 \frac{\ln 20}{\ln 10} \ln 10}=10^{150 \frac{\ln 20}{\ln 10}}$$

Now, $150 \frac{\ln 20}{\ln 10}\approx 195.15$.

|cite|improve this answer
$\endgroup$
2
$\begingroup$

$$\log_{10}20^{150}=150\log_{10}20=150(\log_{10}2+\log_{10}10)\approx 150\cdot 1.30103\approx 195$$

|cite|improve this answer
$\endgroup$
1
$\begingroup$

Hint: $\;2^{10} = 1024 \simeq 1000 = 10^3\,$, so $\,2^{150}=\left(2^{10}\right)^{15} \simeq \left(10^3\right)^{15} = 10^{45}\,$.

|cite|improve this answer
$\endgroup$
0
$\begingroup$

$$ 20^{150} = 2^{150} \times 10^{150} \\ \approx 1.47 \times 10^{45} \times 10^{150} \\ = 1.47 \times 10^{195} $$

alternatively, take the base ten log

$$ \log( 20^{150} ) = 150 \log(20) \approx 195.154 $$

so $$ 20^{150} \approx 10^{ 195.154 } = 10^{0.154} \times 10^{195} \approx 1.47 \times 10^{195} $$

|cite|improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.