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A study of urban density shows that the population density C (in persons //km^(2) ) is related to the distance a (in km ) from the city centre by log_(e)C=log_(e)z-va+ta^(4), where z,v, and t are positive constants. Solve for C as a function of a.

A study of urban density shows that the population density C \mathrm{C} (in persons /km2 / \mathrm{km}^{2} ) is related to the distance a (in km \mathrm{km} ) from the city centre by log⁡eC=log⁡ez−va+ta4 \log _{e} \mathrm{C}=\log _{e} \mathrm{z}-\mathrm{va}+\mathrm{ta}^{4} , where z,v \mathrm{z}, \mathrm{v} , and t \mathrm{t} are positive constants. Solve for C \mathrm{C} as a function of a.

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Q. A study of urban density shows that the population density C \mathrm{C} (in persons /km2 / \mathrm{km}^{2} ) is related to the distance a (in km \mathrm{km} ) from the city centre by log⁡eC=log⁡ez−va+ta4 \log _{e} \mathrm{C}=\log _{e} \mathrm{z}-\mathrm{va}+\mathrm{ta}^{4} , where z,v \mathrm{z}, \mathrm{v} , and t \mathrm{t} are positive constants. Solve for C \mathrm{C} as a function of a.
  1. Write Equation: Write down the given logarithmic equation.\newlineThe given equation is log⁡e(C)=log⁡e(z)−va+ta4\log_e(C) = \log_e(z) - va + ta^4, where zz, vv, and tt are positive constants.
  2. Combine Terms: Use the properties of logarithms to combine the terms on the right side of the equation.\newlineSince log⁡e(z)−va+ta4\log_e(z) - v_a + t a^4 is written as a single logarithm, we can combine the terms using the properties of logarithms. However, we notice that the terms vav_a and ta4t a^4 are not inside a logarithm, which indicates a potential issue with the equation as it stands. We need to correct this before proceeding.

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