Interpret and use a logarithmic model

Using a logarithmic model

A logarithmic model maps an input to an output. Substitute inside the logarithm for a forward calculation, or isolate the logarithm and exponentiate when solving for an input

1. Identify the direction Use substitution when the input is known and isolate the logarithm when the output is known \(\displaystyle Y(t)=c+a\log_e\left(t\right)\)
2. Evaluate an e-power An input e^r makes the natural logarithm equal to r. \(\displaystyle Y\left(e^2\right)=30+4\log_e\left(e^2\right)=30+4(2)\)
3. Isolate before exponentiating For a reverse calculation, remove the additive constant and divide by the logarithmic coefficient before using e. \(\displaystyle Y(t)=30+4\log_e(t)=38\quad\Rightarrow\quad \log_e(t)=2\quad\Rightarrow\quad t=e^2\)
4. Check the model domain The complete logarithm argument must be positive \(\displaystyle \log_e\left(t+b\right)\text{ requires }t+b>0\)

Worked example

Find the response level for the model and input shown

1. \(\displaystyle t=e^2>0\)The stated input satisfies the positive domain condition
2. \(\displaystyle \log_e(t)=\log_e\left(e^2\right)=2\)Evaluate the natural logarithm using the exponent of e
3. \(\displaystyle Y\left(e^2\right)=24-3(2)=18\)Substitute the logarithm value into the model to obtain a response level of 18

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