$2^x=4.019$, solve for x

Question

If I know that $2^b= 4.019$, how do I solve for $b$? I'm sure it's an algebraic method, but not knowing the term makes it impossible to search for.

Answer 1

By definition: $$b^x=y \implies x=\log_b(y)$$ Where $\log_b(y)$ is the logarithm to a base $b$.

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Note that $\log_b(y)$ can also be written as: $$\log_b(y)=\frac{\log_c(y)}{\log_c(b)}$$ Where $c$ is any positive base not equal to $1$. You can choose any value of $c$, one may also choose base $e$ to obtain: $$\log_b(y)=\frac{\ln(y)}{\ln(b)}$$

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Therefore, to answer your question: $$2^x=4.019 \implies x=\log_2(4.019)=\frac{\log_c(4.019)}{\log_c(2)}=\frac{\ln(4.019)}{\ln(2)}\approx 2.00684$$