Logistic Regression - Converting Equations

Question

I am learning logistic regression by watching this video.

How I can reduce eqn?

$$p = \frac{e^{\beta_0 + \beta_1 * age}}{e^{\beta_0 + \beta_1 * age} + 1} $$

to

$$\log_e \frac{p}{1-p} = \beta_0 + \beta_1*age $$

Answer 1

Let $K = e^{\beta_0 + \beta_1 \times age}$. Then

$$p = \frac K{K+1} \leadsto pK+p=K\leadsto p=K-Kp\leadsto\frac p{1-p}=K$$

Finally take $\log_e$ on both sides.

Answer 2

$p = \frac{k}{k+1} \Rightarrow kp+p = k \Rightarrow k = \frac{p}{1-p}$

here $k = e^{\beta_0+\beta_1 age}$. Take $\ln$ on both sides.