# Converting $20^{150}$ into base $10$

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.

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

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

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

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

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

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

alternatively, take the base ten log

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

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