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