Exam 4: Exponential and Logarithmic Functions

Solve the problem. -Suppose that $P$ dollars in principal is invested in an account earning $5.3 \%$ interest compounded continuously. At the end of $3 \mathrm { yr }$ , the amount in the account has earned $\$ 1,378.70$ in interest. Find the original principal. Round to the nearest dollar. (Hint: Use the model $A = P e ^ { r t }$ and substitute $P + 1,378.70$ for $A$ .)

Solve the problem. -Carbon dating determines the approximate age of an object made from materials that were once alive by measuring the remaining percentage of a radioactive isotope, carbon-14. The age is calculated using the formula $A = - \frac { \ln P } { 0.000121 }$ where $P$ is the percentage of remaining carbon-14 (in decimal form). A specimen is determined to be 2,800 years old. What is the percentage of remaining carbon-14? Round to the nearest tenth of a percent.

Write the logarithm as a sum or difference of logarithms. Simplify each term as much as possible. - $\log _ { 2 } \left( \frac { 1 } { 32 } p ^ { 3 } q \right)$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The function defined by $f ( x ) = 1 ^ { x }$ (is/is not) an exponential function.

Solve the equation. - $( \ln x ) ^ { 2 } - \ln x ^ { 5 } = - 6$

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. - $f ( x ) = 2 ^ { x } ; f ( \sqrt { 6 } )$

Solve the problem. -James wants to invest $8,000. He can invest the money at 7.2% simple interest for 30 yr or he can invest at 7% with interest compounded continuously for 30 yr. Which option results in more total Interest?

Simplify the expression without using a calculator. - $\log _ { 6 } \sqrt { \frac { 1 } { 216 } }$

Solve the equation. Write the solution set with the exact values given in terms of natural or common logarithms. Also give approximate solutions to 4 decimal places, if necessary. - $10 ^ { 9 x - 1 } - 8,000 = 110,000$ ; use common logarithms

Simplify the expression. - $\log _ { 2 } 16$

Find the inverse mentally. - $f ( x ) = \sqrt [ 7 ] { 8 x } + 5$

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. - $x ^ { 2 } e ^ { x } = 64 e ^ { x }$

Simplify the expression. - $\log _ { \pi } 1$

Write the domain in interval notation. - $f ( x ) = \ln \left( x ^ { 2 } + 8 x + 12 \right)$

Solve the problem. -The population of a country is modeled by the function $P ( t ) = 15.9 e ^ { 0.01462 t }$ where $P ( t )$ is the population (in millions) $t$ years after January 1, 2000. Use the model to predict the year during which year the population will reach 30 million if this trend continues.

Simplify the expression without using a calculator. - $\log _ { 1 / 2 } \left( \frac { 1 } { 16 } \right)$

Solve the equation. Write the solution set with the exact values given in terms of natural or common logarithms. Also give approximate solutions to 4 decimal places, if necessary. - $4 ^ { t } = 77$ ; use natural logarithms

Write the logarithmic expression as a single logarithm with coefficient 1, and simplify as much as possible. - $7 \log x - \frac { 1 } { 7 } \log y - \frac { 4 } { 7 } \log z$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given $y = \log _ { b } x$ , the value $y$ is called the ـــــــــــ $b$ is called the ــــــــــــــ and $x$ is called the ـــــــــــــــــــ.

Solve the problem. -Suppose $\$ 14,000$ is invested with $4 \%$ interest for 5 yr under the following compounding options. Complete the table. Compounding Option n Value Result a. Daily b. Continuously