Exam 9: Exponential and Logarithmic Functions

The following graph shows how the value of the average car depreciates as a percent of its original value over a 10-year period. It also shows the yearly maintenance costs as a percent of the car s value. When is the car worth half of its purchase price? At the end of the __________ year. When is the car worth 30% of its purchase price? During the __________ year. When do the average yearly maintenance costs surpass the value of the car? During the __________ year.

Compare the graph of $f ( x ) = e ^ { x } + 7$ to the graph of $f ( x ) = e ^ { x }$ .

Find the length of time for $25000 to triple when it is invested at 5% annual interest, compounded continuously. Please round the answer to one decimal place. __________ yr

The graph represents a function. Use the horizontal line test to decide whether the function is one-to-one. $\{ ( 1,4 ) , ( 2,4 ) , ( 3,4 ) , ( 4,4 ) \}$ Answer yes or no .

Solve the equation. Round answer to four decimal places. $5 ^ { x } = 9$

Solve the equation. Give your answer to four decimal places. $13 ^ { x - 1 } = 3$ If there is more than one solution, separate your answers with commas.

The __________ line test can be used to decide whether the graph of a function represents a one-to-one function.

Evaluate: $\ln \sqrt { e }$

Use the properties of logarithms to write the expression $\log _ { 7 } \frac { x y } { 7 }$ as a sum, difference, or multiple of logarithms. Assume x and y are positive numbers.

Solve for x . $\log _ { 4 } x = 0$

An equation with a variable in its exponent, such as $5 ^ { 3 x } = 6$ , is called a(n) __________ equation.

The __________ of f and g , denoted as f+g , is defined by $( f + g ) ( x ) =$ __________ and the __________ of $f$ and $g$ , denoted as $f - g$ , is defined by $( f - g ) ( x ) =$ __________.

Write the logarithm as a sum and/or difference of logarithms of a single quantity. Then simplify if possible. Assume that w and r are positive numbers. $\log _ { 5 } 9$

Use the properties of logarithms to rewrite $\log _ { 2 } \frac { 8 x } { y }$ . Simplify, if possible. Assume x and y are positive numbers.

Let $f ( x ) = 2 x - 9$ and $g ( x ) = x ^ { 2 } + x$ . Find the composition. $( f \circ g ) ( x )$

Write the logarithm as a sum and/or difference of logarithms of a single quantity. Then simplify if possible. Assume that w and r are positive numbers. $\log w ^ { 6 } r ^ { 7 }$

Solve: $\log \left( x ^ { 2 } - 1 \right) - \log ( x + 1 ) = 2$

Assume that $\log _ { b } 4 = 0.6021 , \log _ { b } 7 = 0.8451$ and $\log _ { b } 9 = 0.9542$ . Use these values to evaluate the logarithm. $\log _ { b } 36 =$ __________

Solve: $\log _ { 3 } 7 x - \log _ { 3 } 12 = 1$

What is the range of the function $f ( x ) = e ^ { x }$ ?