Question Type 2: Sketching different logarithmic models for different parameter values using technology Exercises

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

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

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

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

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

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 $b$.

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

A model for acoustic intensity $I$ in decibels as a function of sound pressure $p$ is $I = 20\log_{10}\bigl(\frac{p}{p_{0}}\bigr).$ If $I=60$ when $p=0.02\,$Pa, find the reference pressure $p_{0}$.

The population $P$ of a bacteria colony is modelled by $P(t)=P_{0}+k\ln(1+t),$ where $t$ is in hours. If $P(0)=100$ and $P(3)=160$, find $P_{0}$ and $k$.