IB Mathematics Applications & Interpretation Question Type 2: Sketching Different Logarithmic Models for Different Parameter Values Using Technology Exercises

Given the data points $(1,0)$ and $(10,15)$, fit the model $y=A+B\ln x$. Determine the values of $A$ and $B$.

Solve for $x$: $${\log }_2(x)+{\log }_2(3x-2)=4.$$

A model for acoustic intensity $I$ in decibels as a function of sound pressure $p$ is $$I=20{\log }_{10}\left(\frac{p}{p_0}\right)$$ If $I=60$ when $p=0.02\text{Pa}$, find the reference pressure $p_0$.

Given the logarithmic model $$y=a+b{\log }_{10}(x),$$ which passes through the points $(10,2)$ and $(100,4)$, find the values of $a$ and $b$.

The half-life $T_{1/2}$ of a radioactive substance is related to its decay constant $\lambda$ by $$T_{1/2}=\frac{\ln (2)}{\lambda }.$$ If the half-life is 5 years, solve for $\lambda$.

Find the domain, range, and equation of the vertical asymptote of the function $$y=-2+5\ln (x+4)$$

Find the $x$-intercept and $y$-intercept of the function $y=\ln (5-x).$

Sketch the graphs of $y=\ln (x)$ and $y=1-\ln (x)$ on the same axes. Describe the transformation that maps the graph of $y=\ln (x)$ to the graph of $y=1-\ln (x)$.

Sketch the graphs of $y={\log }_2(x)$ and $y={\log }_2(2x-1)+3$. Indicate the vertical asymptotes, x-intercepts, and describe the transformations applied to the parent function.

The population $P$ of a bacteria colony is modelled by $$P(t)=P_0+k\ln (1+t)$$ where $t$ is in hours. Given that $P(0)=100$ and $P(3)=160$, find $P_0$ and $k$. Give the value of $k$ in simplest exact form.

Sketch the graph of $$y={\log }_{0.5}(x-3)-1$$ indicating the vertical asymptote and two key points on the curve.

Solve the equation $\ln (3x-2)-\ln (x+1)=\ln (2)$.