91Ó°ÊÓ

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ e^{-0.103 x}=7 $$

Determine whether each function is one-to-one. If it is, find the inverse. $$g(x)=(x-4)^{3}$$

Use a calculator to approximate each logarithm to four decimal places. $$\log _{5} 26$$

Solve each equation. Give exact solutions. $$ \log _{6}(4 x+2)=2 $$

Solve each equation. $$ 9^{2 x-8}=27^{x-4} $$

Use a calculator to approximate each logarithm to four decimal places. $$\log 50$$

Solve each equation. $$\log _{12} x=0$$

Solve each equation. Give exact solutions. $$ \log _{2} x+\log _{2}(x+4)=5 $$

A major scientific periodical published an article in 1990 dealing with the problem of global warming. The article was accompanied by a graph that illustrated two possible scenarios. (a) The warming might be modeled by an exponential function of the form \(f(x)=\left(1.046 \times 10^{-38}\right)\left(1.0444^{x}\right)\). (b) The warming might be modeled by a linear function of the form \(g(x)=0.009 x-17.67\). In both cases, \(x\) represents the year, and the function value represents the increase in degrees Celsius due to the warming. Use these functions to approximate the increase in temperature for each year, to the nearest tenth of a degree. $$ 2000 $$

Inverse functions can be used to send and receive coded information. A simple example might use the function \(f(x)=2 x+5 .\) (Note that it is one-to-one.) Suppose that each letter of the alphabet is assigned a numerical value according to its position, as follows. $$\begin{array}{llllllllll}\mathbf{A} & 1 & \mathbf{G} & 7 & \mathbf{L} & 12 & \mathbf{Q} & 17 & \mathbf{V} & 22 \\\\\mathbf{B} & 2 & \mathbf{H} & 8 & \mathbf{M} & 13 & \mathbf{R} & 18 & \mathbf{W} & 23 \\\\\mathbf{C} & 3 & \mathbf{I} & 9 & \mathbf{N} & 14 & \mathbf{S} & 19 & \mathbf{X} & 24 \\\\\mathbf{D} & 4 & \mathbf{J} & 10 & \mathbf{O} & 15 & \mathbf{T} & 20 & \mathbf{Y} & 25 \\\\\mathbf{E} & 5 & \mathbf{K} & 11 & \mathbf{P} & 16 & \mathbf{U} & 21 & \mathbf{Z} & 26 \\\\\mathbf{F} & 6 & & & & & & & &\end{array}$$ Using the function, the word ALGEBRA would be encoded as $$\begin{array}{lllllll}7 & 29 & 19 & 15 & 9 & 41 & 7\end{array}$$ because \(f(\mathrm{~A})=f(1)=2(1)+5=7, \quad f(\mathrm{~L})=f(12)=2(12)+5=29, \quad\) and so on The message would then be decoded using the inverse of \(f,\) which is \(f^{-1}(x)=\frac{x-5}{2}\). $$f^{-1}(7)=\frac{7-5}{2}=1=\mathrm{A}, \quad f^{-1}(29)=\frac{29-5}{2}=12=\mathrm{L}, \quad \text { and so on }$$ You receive the following coded message today. (Read across from left to right.) $$\begin{array}{llllllllllllllll}47 & 95 & 7 & -1 & 43 & 7 & 79 & 43 & -1 & 75 & 55 & 67 & 31 & 71 & 75 & 27\end{array}$$ \(\begin{array}{lllllllllllllll}15 & 23 & 67 & 15 & -1 & 75 & 15 & 71 & 75 & 75 & 27 & 31 & 51 & 23 & 71\end{array}\) $$\begin{array}{llllllllllllll}31 & 51 & 7 & 15 & 71 & 43 & 31 & 7 & 15 & 11 & 3 & 67 & 15 & -1 & 11\end{array}$$ Use the letter/number assignment described on the previous page to decode the message.