Exam 4: Exponential and Logarithmic Functions

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms. $\log _ { 6 } 534$

If $4500 is invested at an interest rate of 6.25% per year, compounded continuously, find the value of the investment after the given number of years. (a) 3 years (b) 6 years (c) 9 years

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms. $\log _ { 2 } 6$

If $6000 is invested in an account for which interest is compounded continuously, find the amount of the investment at the end of 12 years for the following interest rates. (a) 2% (b) 7%

Evaluate the expression.

A sum of $1250 was invested for 8 years, and the interest compounded quarterly. If this sum amounted to $\$ 6583.32$ in the given time, what was the interest rate?

The time required to double the amount of an investment at an interest rate r compounded continuously is given by $t = \frac { \ln 2 } {r }$ Find the time required to double an investment at 2%, 3%, and 4%.

A 12-g sample of radioactive iodine decays in such a way that the mass remaining after t days is given by $m ( t ) = 12 e ^ { - 0087 t }$ , where $m ( t )$ is measured in grams. After how many days is there only 4 g remaining?

Evaluate the expression. $\log _ { \sqrt { 7} } 49$

If $12,000 is invested in an account for which interest is compounded continuously, find the amount of the investment at the end of 12 years for the following interest rates. (a) 2% (b) 7%

If $10,000 is invested at an interest rate of 4% per year, compounded semiannually, find the value of the investment after the given number of years. (a) 5 years (b) 10 years (c) 15 years

Evaluate the expression. $\log _ { 4 } 256$

If after one day a sample of radioactive element decays to $96 \%$ of its original amount, find its half-life. 

For a rat learning to run a maze, the learning curve is given by $S ( t ) = 10 + 8 e ^ { - 0.12 }$ , where $S ( t )$ is the time it takes to run the maze in seconds after $t$ days. Use a graphing calculator or computer to graph of the learning curve and determine after how many days the rat is able to run the maze in $15$ seconds.

The half-life of strontium- $90$ is $25$ years. How long will it take for a $12 \mathrm { mg }$ sample to decay to a mass of $8$ mg? 

The count in a bacteria culture was 320 after one hour and 1000 after five hours. (a) What is the relative growth rate of the bacteria population? (b) What was the initial size of the culture? (c) Find a formula for the number of bacteria $n ( t )$ after $t$ hours. (d) Find the number of bacteria after $8$ hours. (e) When will the number of bacteria be $7000 ?$

(a)Find the inverse function. (b) What is the domain of the inverse function? $f ( x ) = \frac { 5 ^ { 2 x } } { 5 - 5 ^ { 2 x } }$

Radium- $221$ has a half-life of $30 \mathrm {~s}$ . Suppose we have a $300 \mathrm {~g}$ sample. (a) Find a formula for the mass remaining after $t$ seconds. (b) How much of the sample remains after $200$ seconds? (c) After how long will only $20 \mathrm {~g}$ remain?

Use the Laws of Logarithms to rewrite the expression $\ln \left( \frac { x ^ { 3 } } { y ^ { 3 } \sqrt [ 3 ] { z } } \right)$ in a form with no logarithm of a product, quotient, root, or powers.  

Use the Laws of Logarithms to rewrite the expression $\log _ { 3 } \left( \frac { 1 ^ { 5 x - 1 } } { x ( x - 1 ) ( \sqrt [ 3 ] { x - 1 } ) } \right)$ in a form with no logarithm of a product, quotient, root, or powers.