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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) \(2=\log (x-8)\)

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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?

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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 \(24 \mathrm{mm}\)

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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.

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polarization, property of certain electromagnetic radiations in which the direction and magnitude of the vibrating electric field are related in a specified way.

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 \(y=\log _{2} 8 x\).

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Light waves are transverse: that is, the vibrating electric vector associated with each wave is perpendicular to the direction of propagation. A beam of unpolarized light consists of waves moving in the same direction with their electric vectors pointed in random orientations about the axis of propagation. Plane polarized light consists of waves in which the direction of vibration is the same for all waves. In circular polarization the electric vector rotates about the direction of propagation as the wave progresses. Light may be polarized by reflection or by passing it through filters, such as certain crystals, that transmit vibration in one plane but not in others.

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) \(\frac{\log x}{3}+\frac{\log y}{3}\)