Write each expression as a single logarithm in simplest form. a) \(\log _{0} x-\log _{9} y+4 \log _{9} z\) b) \(\frac{\log _{3} x}{2}-2 \log _{3} y\) c) \(\log _{6} x-\frac{1}{5}\left(\log _{6} x+2 \log _{6} y\right)\) d) \(\frac{\log x}{3}+\frac{\log y}{3}\)

Vertical translationequation

Vertical translationmath

To obtain the graph of \(y=\log _{2} 8 x,\) you can either stretch or translate the graph of \(y=\log _{2} x\). a) Describe the stretch you need to apply to the graph of \(y=\log _{z} x\) to result in the graph of \(y=\log _{2} 8 x\). b) Describe the translation you need to apply to the graph of \(y=\log _{2} x\) to result in the graph of \(y=\log _{2} 8 x\).

Small animal characters in animated features are often portrayed with big endearing eyes. In reality, the eye size of many vertebrates is related to body mass by the logarithmic equation \(\log E=\log 10.61+0.1964 \log m,\) where \(E\) is the eye axial length, in millimetres, and \(m\) is the body mass, in kilograms. To the nearest kilogram, predict the mass of a mountain goat with an eye axial length of \(24 \mathrm{mm}\)

Vertical translationand horizontaltranslation

Solve. Give exact answers. a) \(15=12+\log x\) b) \(\log _{5}(2 x-3)=2\) c) \(4 \log _{3} x=\log _{3} 81\) d) \(2=\log (x-8)\)

The largest lake lying entirely within Canada is Great Bear Lake, in the Northwest Territories. On a summer day, divers find that the light intensity is reduced by \(4 \%\) for every meter below the water surface. To the nearest tenth of a meter, at what depth is the light intensity \(25 \%\) of the intensity at the surface?

a) Only a vertical translation has been applied to the graph of \(y=\log _{3} x\) so that the graph of the transformed image passes through the point (9, - 4). Determine the equation of the transformed image. b) Only a horizontal stretch has been applied to the graph of \(y=\log _{2} x\) so that the graph of the transformed image passes through the point \((8,1) .\) Determine the equation of the transformed image.