Question

Convert the following expressions to the indicated base. \(\log _{2}\left(x^{2}+1\right)\) using base \(e\)

Short answer

Question: Convert the following expression from base 2 to base e: \(\log_{2}\left(x^{2}+1\right)\) Answer: \(\frac{\ln\left(x^{2}+1\right)}{\ln(2)}\)

Step-by-step solution

Identify the given expression

The given expression is: \(\log_{2}\left(x^{2}+1\right)\)

Apply the change of base formula

We have to convert the expression using base e. So, we have to apply the change of base formula as follows: $$\log_{2}\left(x^{2}+1\right) = \frac{\log_{e}\left(x^{2}+1\right)}{\log_{e}(2)}$$

Simplify the expression

We can rewrite the expression in a more standard form using the natural logarithm notation for base e: $$\frac{\ln\left(x^{2}+1\right)}{\ln(2)}$$ So, the given expression \(\log_{2}\left(x^{2}+1\right)\) converted to base e is equal to \(\frac{\ln\left(x^{2}+1\right)}{\ln(2)}\).