91Ó°ÊÓ

Research the history of logarithms including the origin of the word 'logarithm' itself. Why is the abbreviation of natural log 'ln' and not 'nl'?

Find the domain of the function. \(f(x)=\log _{7}(4 x+8)\)

Find the domain of the function. \(f(x)=\ln (4 x-20)+\ln \left(x^{2}+9 x+18\right)\)

Sketch the graph of \(y=g(x)\) by starting with the graph of \(y=f(x)\) and using transformations. Track at least three points of your choice and the horizontal asymptote through the transformations. State the domain and range of \(g\). . \(f(x)=10^{x}, g(x)=10^{\frac{x+1}{2}}-20\)

Sketch the graph of \(y=g(x)\) by starting with the graph of \(y=f(x)\) and using transformations. Track at least three points of your choice and the vertical asymptote through the transformations. State the domain and range of \(g\). \(f(x)=\log _{2}(x), g(x)=\log _{2}(x+1)\)

We introduce three widely used measurement scales which involve common logarithms: the Richter scale, the decibel scale and the pH scale. The computations involved in all three scales are nearly identical so pay attention to the subtle differences. While the decibel scale can be used in many disciplines, \(^{13}\) we shall restrict our attention to its use in acoustics, specifically its use in measuring the intensity level of sound. \(^{14}\) The Sound Intensity Level \(L\) (measured in decibels) of a sound intensity \(I\) (measured in watts per square meter) is given by $$L(I)=10 \log \left(\frac{I}{10^{-12}}\right)$$ Like the Richter scale, this scale compares \(I\) to baseline: \(10^{-12} \frac{W}{m^{2}}\) is the threshold of human hearing. (a) Compute \(L\left(10^{-6}\right)\). (b) Damage to your hearing can start with short term exposure to sound levels around 115 decibels. What intensity \(I\) is needed to produce this level? (c) Compute \(L(1)\). How does this compare with the threshold of pain which is around 140 decibels?