Exam 12: Logarithmic and Exponential Functions

Solve. -The half-life of silicon-32 is 710 years. If 100 grams is present now, how much will be present in 900 years? (Round your answer to three decimal places.)

Rewrite in logarithmic form. - $3 ^ { x } = 60$

Evaluate the given function. - $f ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x } , f ( - 3 )$

Solve. -The number of books in a small library increases according to the function B = 7300e0.05t, where t is measured in years. How many books will the library have after 9 years?

Solve the problem. -A certain country's population $\mathrm { P } ( \mathrm { t } )$ , in millions, $\mathrm { t }$ years after 1980 can be approximated by $\mathrm { P } ( \mathrm { t } ) = 3.495 ( 1.016 ) ^ { \mathrm { t } } \text {. }$ In what year will the country's population reach 6 million?

Find all intercepts for the given function. Round to the nearest tenth if necessary. - $f ( x ) = e ^ { ( x + 2 ) } - 6$

Determine the equation of the horizontal asymptote for the graph of this function, and state the domain and range of this function. - $f ( x ) = e ^ { x - 1 } + 1$

Solve the problem. -The hydrogen ion concentration of a substance $\left[ \mathrm { H } ^ { + } \right]$ is about $5.2 \times 10 ^ { - 7 }$ moles per liter. Find the $\mathrm { pH }$ . Round to the nearest tenth. Use the formula $\mathrm { pH } = - \log \left[ \mathrm { H } ^ { + } \right]$ .

Rewrite in logarithmic form. -The population growth of an animal species is described by F(t) = 600 + 60 log3 (2t + 1) where t is the number of months since the species was introduced. Find the population of this species in an Area 40 month(s) after the species is introduced.

Compute the compound interest. -$5000 is invested at 9% compounded quarterly. In how many years will the account have grown to $10,500? Round to the nearest tenth of a year.

Rewrite as a single logarithm. Assume all variables represent positive real numbers. - $4 \log ( x + 3 ) - 9 \log \left( x ^ { 2 } + 4 \right) + \frac { 1 } { 3 } \log y$

Solve the problem. -Use the formula $L = 10 \cdot \log \frac { 1 } { \mathrm { I } _ { 0 } }$ , where the loudness of a sound in decibels is determined by $\mathrm { I }$ , the number of watts per square meter produced by the sound wave, and I $0 = 10 ^ { - 12 }$ watts per square meter. A certain noise produces $9.92 \times 10 ^ { - 5 }$ watts per square meter of power. What is the decibel level of this noise?

Solve the problem. -Use the formula $L = 10 \cdot \log \frac { \mathrm { I } } { \mathrm { I } }$ , where the loudness of a sound in decibels is determined by I, the number of watts per square meter produced by the sound wave, and I $0 = 10 ^ { - 12 }$ watts per square meter. What is the intensity of a noise measured at $56 \mathrm { db }$ ?

Solve the problem. -Approximately one-fourth of all glass bottles distributed will be recycled each year. A beverage company distributes 380,000 bottles. The number still in use after t years is given by the function $\mathrm { N } ( \mathrm { t } ) = 380,000 \left( \frac { 1 } { 4 } \right) ^ { \mathrm { t } }$ . After how many years will 3000 bottles still be in use? Round your answer to the nearest tenth.

Rewrite using the power and product rules. Assume all variables represent positive real numbers. - $\log _ { a } 7 x ^ { 2 } y z ^ { 3 }$

Rewrite using the power rule. Assume all variables represent positive real numbers. - $5 \log _ { b } m$

Expand. Assume that all variables represent positive real numbers. - $\log _ { 11 } \left( \frac { 5 \sqrt { m } } { n } \right)$

Evaluate the given function. - $f ( x ) = 9 \log _ { 2 } ( 3 x ) , \left( \frac { 8 } { 3 } \right)$

Rewrite in logarithmic form. -The number of visitors to a tourist attraction (for the first few years after its opening) can be approximated by $V ( x ) = 50 + 10 \log _ { 2 } x$

Evaluate the given function. - $f ( x ) = 2 ^ { - x } , f ( 9 )$