Exam 4: Exponential and Logarithmic Functions

Choose the one alternative that best completes the statement or answers the question. Solve for the indicated variable. - $\mathrm { pH } = - \log \left( \mathrm { H } ^ { + } \right)$ for $\mathrm { H } ^ { + }$

Use the quotient property of logarithms to write the logarithm as a difference of logarithms. Then simplify if possible. - $\ln \left( \frac { e } { 18 } \right)$

Use the change-of-base and a calculator to approximate the logarithm to 4 decimal places. - $\log _ { 9 } 0.6$

Solve the problem. - $f ( x ) = \sqrt { x + 3 }$ a. Graph $f ( x )$ b. Write an equation for $f ^ { - 1 } ( x )$ c. Write the domain of $f ^ { - 1 }$ in interval notation

Simplify the expression without using a calculator. - $\log _ { 2 } \sqrt [ 5 ] { 2 }$

Choose the one alternative that best completes the statement or answers the question. A relation in x and y is given. Determine if the relation defines y as a one-to-one function of x. - x y 1.5 1.12 5.8 -0.34 -6.2 -2.56 4.3 -0.66

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given y = logb x, if b > 1, then the graph of the function is a(n) (increasing/decreasing) logarithmic function. If 0 < b < 1, then the graph is (increasing/decreasing).

Solve the equation. - $\log x + 7 \sqrt { \log x } - 18 = 0$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given $y = \log x$ , the base is understood to be . Given $y = \ln x$ , the base is understood to be .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The change-of-base formula is often used to convert a logarithm to a ratio of logarithms with base or base so that a calculator can be used to approximate the logarithm.

Solve the problem. -Use the graph of $y = 5 ^ { x }$ to graph the function. Write the domain and range in interval notation. $f ( x ) = 5 ^ { x } - 5$

Write the logarithm as a sum or difference of logarithms. Simplify each term as much as possible. - $\log _ { 8 } \sqrt [ 3 ] { y _ { \sqrt { 8 } } }$

Solve the equation. - $\log _ { 3 } ( n - 3 ) + \log _ { 3 } ( n + 5 ) = 2$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If $k > 0$ , the equation $y = y _ { 0 } e ^ { k t }$ is a model for exponential (growth/decay), whereas if $k < 0$ , the equation is a model for exponential (growth/decay).

Solve the problem. -The local magnitude $M _ { L }$ (on the Richter scale) of an earthquake of intensity $I$ is given by $M _ { L } = \log \left( \frac { I } { I _ { 0 } } \right)$ where $I _ { 0 }$ is a minimum reference intensity of a "zero-level" earthquake against which the intensities of other earthquakes may be compared. How many times more intense is an earthquake of magnitude $5.8$ than an earthquake of magnitude $2.9$ ? Round to the nearest whole number.

Solve the problem. -After a new product is launched the cumulative sales $S ( t )$ (in $\$ 1000 ) t$ weeks after launch is given by $S ( t ) = \frac { 54 } { 1 + 11 e ^ { - 0.32 t } }$ Determine the amount of time for the cumulative sales to reach $\$ 48,000$ . Round to the nearest week.

Write the logarithm as a sum or difference of logarithms. Simplify each term as much as possible. - $\ln \left[ \frac { 5 x \left( x ^ { 9 } + 4 \right) ^ { 9 } } { \sqrt [ 9 ] { 3 + 5 x } } \right]$

Solve the problem. -Eight million E. coli bacteria are present in a laboratory culture. An antibacterial agent is introduced and the population of bacteria $P ( t )$ decreases by half every $5 \mathrm { hr }$ . The population can be represented by $P ( t ) = 8,000,000 \left( \frac { 1 } { 2 } \right) ^ { t / 5 }$ . Convert this to an exponential function using base $e$ .

Choose the one alternative that best completes the statement or answers the question. Evaluate the function at the given value of x. Round to 4 decimal places if necessary. - $f ( x ) = 4 ^ { x } ; f ( 4.2 )$

Write as the sum or difference of logarithms and fully simplify, if possible. Assume the variable represents a positive real number. - $\log \left( \frac { \sqrt [ 3 ] { a b } } { c ^ { 5 } } \right)$