Semi-log Plots

What AP Precalc asks here

If $y = a \cdot b^x$ then $\log y = \log a + x \log b$ — a linear function of $x$. So plotting $\log y$ vs $x$ (a semi-log plot) turns exponential data into a straight line, with slope $\log b$ and intercept $\log a$. Similarly, $y = a x^p$ becomes linear on log-log axes with slope $p$. Reading the slope is what recovers $b$ or $p$ — but you must back-convert: $b = 10^{\text{slope}}$ for common log, $b = e^{\text{slope}}$ for natural log.

Compute $\log y$ for each data point. If the values rise by a constant step as $x$ rises by a constant step, the data is exponential.

Recognize that the y-intercept on $\log y$ gives $\log a$ (so $a = 10^{\text{intercept}}$) and the slope gives $\log b$ (so $b = 10^{\text{slope}}$).

If $\log y$ vs $\log x$ is linear with slope $p$ and passes through the origin (in log-log coords), then $y = x^p$.