91Ó°ÊÓ

Simplify the following expressions: (a) \(\log _{4} 16^{x}\) (b) \(\log _{2} 16^{x}\) (c) \(\log _{3} 27^{x}\) (d) \(\log _{1 / 2} 4^{x}\) (e) \(\log _{1 / 2} 8^{-x}\) (f) \(\log _{3} 9^{-x}\)

Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship either in log or semilog transformed coordinates so that a straight line results. $$ y=5^{x} ; \text { base } 5 $$

Simplify the following expressions: (a) \(-\log _{3} \frac{1}{4}\) (b) \(\log \left(\frac{x^{3}-x}{x-1}\right)\) (c) \(\ln \left(e^{x-2}\right)\)

Simplify the following expressions: (a) \(\ln x^{2}+\ln x^{-1}\) (b) \(\ln x^{4}-\frac{1}{3} \ln x^{-2}\) (c) \(\ln \left(x^{2}-1\right)-\ln (x+1)\) (d) \(\frac{1}{2} \ln x^{-1}+\ln x^{-3}\)

Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship either in log or semilog transformed coordinates so that a straight line results. $$ y=2^{-x} ; \text { base } 2 $$

Solve for \(x\). (a) \(e^{3 x-1}=2\) (b) \(e^{-2 x}=10\) (c) \(e^{x^{2}-1}=10\)

Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship either in log or semilog transformed coordinates so that a straight line results. $$ y=3 e^{-2 x} ; \text { base } 3 $$

Solve for \(x\). (a) \(5^{x}=625\) (b) \(4^{4 x}=256\) (c) \(10^{2 x}=0.0001\)

Simplify the following expressions: (a) \(e^{3 \ln x}\) (b) \(e^{-\ln \left(x^{2}+1\right)}\) (c) \(e^{-2 \ln (1 / x)}\) (d) \(e^{-2 \ln x}\)

Suppose that \(N(t)\) denotes a population size at time \(t\) and satisfies the equation $$ N(t)=2 e^{3 t} \quad \text { for } t \geq 0 $$ (a) If you graph \(N(t)\) as a function of \(t\) on a semilog plot, a straight line results. Explain why. (b) Graph \(N(t)\) as a function of \(t\) on a semilog plot, and determine the slope of the resulting straight line.