Simple logarithmic equation$ (\log 2^x = 180)$

Question

I've been trying to wrap my head around a simple logarithmic equation.

So, there are many ways to represent it:

• $10^{180} = 2^x$

• $\log(2^x) = 180$

• $5^{180} = 2^{(x-180)}$

(If you could not use Napierian logarithm while resolving - or at least explain it when used - I would be thankful)

Answer 1

We have that for injectivity

$$10^{180}=2^x$$

$$\log \left(10^{180}\right)=\log (2^x) $$

$$180 \log 10 =x\log 2$$

$$x=\frac{180 \log 10}{\log 2}$$

that is

• in base $10 \implies x=\frac{180 }{\log 2}$

• in base $2 \implies x= 180 \log 10$

Answer 2

$$\log x^2=180$$ $$10^{\log x^2}=10^{180}$$ $$ x^2=10^{180}$$ Thus $$ x=\pm 10^{90}$$

Answer 3

I found the solution:

• 10^180 = 2^x
• log 10^180 = log 2^x
• 180(log 10) = x(log 2)

• 180*1 = x(log 2)

• x = 180/log 2

• x = 597.94705708