Exam 4: Exponential and Logarithmic Functions

A substance has a hydrogen ion concentration of $\left[ \mathrm { H } ^ { \cdot } \right] = 3.7 \times 10 ^ { - 6 }$ M. Find the pH of the substance.

The height $h$ from ground level to a hanging cable suspended between two utility poles can be described by the function $h ( x ) = \frac { a } { 2 } \left( e ^ { x / a } + e ^ { - x / a } \right)$ , where $x$ is the horizontal distance in feet, measured from the origin, and $a$ is the lowest point of the hanging cable. (a) Find the height of a cable, correct to two decimal places, $10$ feet to the right of its origin and its lowest point $10$ feet above ground level. (b) Find the height of a cable, correct to two decimal places, $25$ feet to the left of its origin and its lowest point $35$ feet above ground level.

Graph the function, not by plotting points, but by starting from the graph of $y = e ^ { x }$ . State the domain, range, and asymptote. $h ( x ) = e ^ { x + 1 } - 5$

Solve the equation $\left( x ^ {1 } \right) ^ { \log ^ { 2 } } = 100 x ^ { 2 }$ .

Use a calculator to evaluate the expression $\ln ( 4 - \sqrt { 10 } )$ correct to four decimal places.

Express the equation $\log _ { 32 } 8 = \frac { 3 } { 5 }$ in exponential form.

Assume that the rat population in a certain urban area behaves according to the logistic growth function $P ( t ) = \frac { 20 } { 1 + 9 e ^ { - 0.31 t } }$ where $P$ is the number of rats in the hundreds, and $t$ is measured in months.(a) Find the initial rat population.(b) Find the rat population after $1$ month.(c) After how many months will the rat population triple from its initial size?

Evaluate the expression. $\log \sqrt { 0.0001 }$

Find the solution of the equation $\left( \frac { 3 } { 5 } \right) ^ { 7 x } = 12$ correct to four decimal places.

If the pH readings of an unknown substance is pH $= 8.3$ , find the hydrogen ion concentration.

Graph the function, not by plotting points, but by starting from the graph of $y = e ^ { x }$ . State the domain, range, and asymptote. $h ( x ) = e ^ { x - 3 } + 5$

Evaluate the expression. $\log _ { 12 } 3 + \log _ { 12 } 48$

A sum of $\$ 2000$ was invested for four years and the interest was compounded quarterly. If this sum amounted to $\$ 2588.44$ after the given time, what was the interest rate?

Graph the function $f ( x ) = 3 + \ln ( 2 - x )$ not by plotting points but by applying your knowledge of the shape of the log function. State domain, range, and asymptote.

Graph the function $y = - 3 + 2 \left( 1 + 3 ^ { x } \right)$ , not by plotting points but by applying your knowledge of the general shape of graphs of the form $a ^ { x }$ . State the domain, range, and asymptote of the function.

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $h ( x ) = 3 e ^ { x } , \quad h ( 3 ) , h ( 0.23 ) , h ( 1 ) , h ( - 2 )$

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $f ( x ) = 2 ^ { x } , \quad f ( 0.5 ) , f ( \sqrt { 2 } ) , f ( \pi ) , f ( 1 / 3 )$

Graph the function $h ( x ) = \log _ { 10 } | x + 1 |$ , not by plotting points but by applying your knowledge of the shape of the log function. State domain, range, and asymptote.  

Use a calculator to evaluate the expression $\log _ { 10 } \left( \frac { 4 } { 5 } \right)$ correct to four decimal places.

The air resistance of a free falling object is proportional to its velocity. Use the equation of downward velocity given by $v ( t ) = 32 \left( e ^ { - p t } - 1 \right)$ , where $t$ is time and $v$ is measured in ft/s, to find the following. (a) The initial velocity of the object. (b) The velocity of the object after $2$ s and $10$ s, if $p = 0.25$ . (c) The object's maximum or terminal velocity, and a graph of the object's velocity.