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Uninhibited growth can be modeled by exponential functions other than \(A(t)=A_{0} e^{k t} .\) For example, if an initial population \(P_{0}\) requires \(n\) units of time to double, then the function \(P(t)=P_{0} \cdot 2^{t / n}\) models the size of the population at time t. Likewise, a population requiring \(n\) units of time to triple can be modeled by \(P(t)=P_{0} \cdot 3^{t / n}\). An insect population grows exponentially. (a) If the population triples in 20 days, and 50 insects are present initially, write an exponential function of the form \(P(t)=P_{0} \cdot 3^{t / n}\) that models the population. (b) What will the population be in 47 days? (c) When will the population reach \(700 ?\) (d) Express the model from part (a) in the form \(A(t)=A_{0} e^{k t}\)

Uninhibited growth can be modeled by exponential functions other than \(A(t)=A_{0} e^{k t} .\) For example, if an initial population \(P_{0}\) requires \(n\) units of time to double, then the function \(P(t)=P_{0} \cdot 2^{t / n}\) models the size of the population at time t. Likewise, a population requiring \(n\) units of time to triple can be modeled by \(P(t)=P_{0} \cdot 3^{t / n}\). The population of a town is growing exponentially. (a) If its population doubled in size over an 8 -year period and the current population is 25,000 , write an exponential function of the form \(P(t)=P_{0} \cdot 2^{t / n}\) that models the population. (b) What will the population be in 3 years? (c) When will the population reach \(80,000 ?\) (d) Express the model from part (a) in the form \(A(t)=A_{0} e^{k t}\).

Suppose that \(\ln 2=a\) and \(\ln 3=b .\) Use properties of logarithms to write each logarithm in terms of a and \(b\). \(\ln 1.5\)

What rate of interest compounded quarterly will yield an effective interest rate of \(7 \%\) ?

If Tanisha has \(\$ 100\) to invest at \(4 \%\) per annum compounded monthly, how long will it be before she has \(\$ 150 ?\) If the compounding is continuous, how long will it be?

If Angela has $$ 100\( to invest at \)2.5 \%\( per annum compounded monthly, how long will it be before she has \)\$ 175 ?$ If the compounding is continuous, how long will it be?

Write each expression as a sum and/or difference of logarithms. Express powers as factors. \(\log _{7} x^{5}\)

Verify that the functions \(f\) and g are inverses of each other by showing that \(f(g(x))=x\) and \(g(f(x))=x\). Give any values of x that need to be excluded from the domain of \(f\) and the domain of g. $$ f(x)=\frac{2 x+3}{x+4} ; \quad g(x)=\frac{4 x-3}{2-x} $$

Verify that the functions \(f\) and g are inverses of each other by showing that \(f(g(x))=x\) and \(g(f(x))=x\). Give any values of x that need to be excluded from the domain of \(f\) and the domain of g. $$ f(x)=\frac{x-5}{2 x+3} ; \quad g(x)=\frac{3 x+5}{1-2 x} $$

What will a $$ 90,000\( condominium cost 5 years from now if the price appreciation for condos over that period averages \)3 \%$ compounded annually?