Exam 4: Logarithm Functions

Which of the following functions y = f(x) satisfy y'= 32y, f (0) = $\frac { 1 } { 2 }$ ? (I) y = 32 $e ^ { 1 / 2 x }$ (II) y = $\mathrm { e } ^ { 16 x }$ (III) y = $\frac { 1 } { 2 }$ $e ^ { 32 x }$ (IV) y = $\frac { 1 } { 2 }$ $x ^ { 32 }$

$e ^ { x - 1 }$ + 6 = 4 $e ^ { x - 1 }$ Enter your answer exactly as just a ± ln b.

Find k such that $3 ^{- x / 2}$ = $\mathrm { e } ^ { \mathrm { kx } }$ for all x.

f(x) = 4 $e ^ { 3 x }$ Enter your answer exactly as just a $\mathrm { e } ^ { \mathrm { b } }$ .

ln $\mathrm { e } ^ { 2 x }$ - ln $\mathrm { e } ^ { - \mathrm { x } / 2 }$ Enter just a standard polynomial in x with any fractions reduced of form $\frac { a } { b }$ .

$e ^ { \ln x - 2 \ln y }$ Enter just a standard polynomial in x.

Which of the following is the largest number?

Use logarithmic differentiation to differentiate. - $4 ^ { x }$ ∙ $5 ^ { x }$ ∙ $6 x ^ { 3 }$ at x = 1 Enter your answer exactly as just a(b ± ln c) where a, b, and c are integers.

Find the equation of the tangent line to the curve y = $\frac { 1 } { x }$ $e ^ { x }$ at (1, e) in the simplest form.

If $9t$ ∙ $34 t$ ∙ $9 - 2 \mathrm { t }$ = $\sqrt { 3 }$ , find t.

Differentiate. -(1 - 4 $e ^ { x }$ ) $x ^ { 2 }$

Solve for x: $24 - x$ ∙ $2 ^ { 8 } + 2 x$ = 64. Enter your answer exactly as x = a (with a an integer).

ln(2 $x ^ { 2 }$ + 1)Enter your answer exactly as $\frac { P ( x ) } { Q ( x ) }$ where P and Q are polynomials in x in standard form.

Solve for x: ( $5 ^ { x }$ ∙ 25 $)^2$ = 125 ∙ $\left( \frac { 1 } { 25 } \right) ^ { x }$ . Enter your answer exactly as x = $\frac { a } { b }$ in lowest terms.

Choose the one alternative that best completes the statement or answers the question. - $3 ^ { 2 } p$ ∙ $9 \mathrm { p }$ ∙ $4 \mathrm { P }$

$e ^ { \ln ( 3 x ) }$ - ln( $\mathrm { e } ^ { 4 }$ ) = 1 Enter your answer exactly as x = $\frac { a } { b }$ .

If $\left( \frac { 1 } { 4 } \right) ^ { 3 x + 1 }$ = $2 ^ { 6 } - 2 x$ , find x.

ln $\frac { 1 + x ^ { 2 } } { 2 x + 5 }$ at x = 1 Enter your answer as just a reduced fraction $\frac { a } { b }$ .

$\mathrm { e } ^ { \mathrm { x } }+ 2 \ln \mathrm { x }$ Enter your answer exactly as $a^ b$ $\mathrm { e } ^ { \mathrm { C } }$ .

If ln x = -3.6, write x using the exponential function.