Exam 4: Exponential and Logarithmic Functions

Solve the problem. -A city is growing at the rate of 0.5% annually. If there were 3,110,000 residents in the city in 1,995, find how many (to the nearest ten-thousand) were living in that city in 2000. Use $y = 3,110,000 ( 2.7 ) ^ { 0.005 t }$

The function f is one-to-one. Find its inverse. - $f ( x ) = x ^ { 3 } + 4$

Use a calculator to find the natural logarithm correct to four decimal places. - $\ln \left( 1.65 \cdot e ^ { - 6 } \right)$

Solve the problem. -A thermometer reading 84°F is placed inside a cold storage room with a constant temperature of 38°F. If the thermometer reads 77°F in 11 minutes, how long before it reaches 56°F? Assume the cooling follows Newton's Law Of Cooling: $\mathrm { U } = \mathrm { T } + \left( \mathrm { U } _ { \mathrm { O } } - \mathrm { T } \right) \mathrm { e } ^ { \mathrm { kt } }$ (Round your answer to the nearest whole minute.)

Graph the function. - $f(x)=2-\ln (x+4)$ A) B) C) D)

For the given functions f and g, find the requested composite function. - $f ( x ) = \frac { x - 2 } { 3 } , g ( x ) = 3 x + 2 ; \quad$ Find $( g \circ f ) ( x )$

Solve the equation. - $9 ^ { 2 x } \cdot 27 ^{( 3 - x )} = \frac { 1 } { 9 }$

Solve the given exponential equation. Round answer to three decimal places. - $2 ^ { x } - 4 x + 2 = 0$

Solve the problem. -During 1991, 200,000 people visited Rave Amusement Park. During 1997, the number had grown to 834,000. If the number of visitors to the park obeys the law of uninhibited growth, find the exponential growth function That models this data. A) $f ( t ) = 634,000 e ^ { 0.248 t }$ B) $f ( t ) = 200,000 e ^ { 0.238 t }$ C) $\mathrm { f } ( \mathrm { t } ) = 634,000 \mathrm { e } ^ { 0.238 \mathrm { t } }$ D) $f ( t ) = 200,000 e ^ { 0.248 t }$

The graph of a one-to-one function f is given. Draw the graph of the inverse function f-1 as a dashed line or curve. - $f ( x ) = x ^ { 3 } + 4$ A) B)

Graph the function. - $f(x)=-1+e^{x}$ A) B) C) D)

Decide whether the composite functions, f $f \circ g$ nd $\mathbf { g } \circ \mathrm { f }$ f, are equal to x. - $f ( x ) = \sqrt { x } , g ( x ) = x ^ { 2 }$

Use transformations to graph the function. - $\text { Use the graph of } \log _ { 5 } x \text { to obtain the graph of } f ( x ) = - 2 \log _ { 5 } x \text {. }$ A) B) C) D)

The loudness of a sound of intensity x, measured in watts per square meter, is defined as L( $L ( x ) = \log \left( \frac { x } { x _ { 0 } } \right) , \text { where } x _ { 0 } = 10 ^ { - 3 }$ -At a recent Phish rock concert, sound intensity reached a level of 0.50 watt per square meter. To the nearest whole number, calculate the loudness of this sound in decibels.

Solve the equation. - $\log _ { 5 } ( x + 4 ) = 3$

Write as the sum and/or difference of logarithms. Express powers as factors. - $\log _ { b } \sqrt [ 3 ] { \frac { x ^ { 5 } y ^ { 8 } } { z ^ { 2 } } }$

Solve the problem. -How long does it take $1700 to double if it is invested at 5% interest, compounded monthly? Round your answer to the nearest tenth.

Solve the problem. -The value of a particular investment follows a pattern of exponential growth. In the year 2000, you invested money in a money market account. The value of your investment t years after 2000 is given by the exponential Growth model A $A = 8,500 e ^ { 0.065 t }$ . By what percentage is the account increasing each year?

Solve the problem. -The rates of death (in number of deaths per 100,000 population) for 20-24 year olds in the United States between 1985-1993 are given below. (Source: NCHS Data Warehouse) Year Rate of Death 1985 134.9 1987 154.7 1989 162.9 1991 174.5 1993 182.2 A logarithmic equation that models this data i $y = 57.76 + 48.56 \ln x$ 56 ln x where x represents the number of years since 1980 and y represents the rate of death in that year. Use this equation to predict the year in which the rate of death for 20-24 year olds first exceeds 200.

Graph the function using a graphing utility and the Change-of-Base Formula. - y=x A) B) C) D)