Exam 4: Logarithm Functions

ln( $e ^ { x }$ + $\mathrm { e } ^ { - \mathrm { x } }$ ) at x = 0 Enter just an integer.

$\frac { 1 } { 3 }$ ln 27 - 2 ln 4 + ln 3 + $( \ln 2 ) ^ { 2 }$ - $\mathrm { e } ^ { \ln 6 + 1 / 4 \ln 81 }$ Enter your answer exactly as just $( \ln a ) ^ { n } + \ln \left( \frac { b } { c } \right) - d$ .

Solve for x. - $e ^ { x ^ { 2 } - 4 }$ = 3

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

If $5^t$ + $5^t$ + $5^t$ = 75, find t.

Choose the one alternative that best completes the statement or answers the question. - $5 ^ { 2 } \mathrm { p }$ ∙ $4 \mathrm { P }$ ∙ $9 \mathrm { p } / 2$

Simplify - $\left( \frac { 1 } { 9 } \right) ^ { 27 x }$ ∙ $\left( \frac { 1 } { 3 } \right) ^ { 48 x }$ ∙ $\left( \frac { 1 } { 81 } \right) ^ { 9 x }$ Enter your answer as $3 ^a$ .

Solve for x: $5 ^ { x }$ ∙ $5^{2 x}$ ∙ $53 x$ = 25. Enter your answer exactly as x = $\frac { a } { b }$ in lowest terms.

Simplify. - $\left( \mathrm { e } ^ { 3 x } \right.$ $)^2$ ∙ $\mathrm { e } ^ { - \mathrm { x } }$ Enter your answer as $e ^ { a }$ .

The expression may be factored as shown. Find the missing factor. $2 ^ { 3 x / 2 } + 2 ^ { - x / 2 } = \sqrt { 2 ^ { - x } }$ ( )Enter your answer as $2^ a$ + b.

$( \ln x ) ^ { 5 }$ Enter your answer as $\frac { P ( \ln x ) } { Q ( x ) }$ where P is a polynomial in lnx and Q is a polynomial in x.

Simplify - $\left( \frac { 1 } { 3 x } \right) ^ { 3 x }$ Enter your answer exactly in the form $3 \mathrm { P } ( \mathrm { x } )$ , where P(x) is a polynomial.

Which of the following functions satisfy the differential equation y' = -8y? (I) y = - $e ^ { - 8 x }$ (II) y = $e ^ { - 4 x }$ + 3 (III) y = 5 $e ^ { - 4 x }$ (IV) y = 6 $e ^ { - 8 x }$

ln(1 - 2x) = 2 ln(1 - x)Enter just a real number.

f(x) = $e ^ { 1 / x }$ Enter your answer exactly as just $a ^b$ $e ^ { c ^ { d } }$ .

Simplify - $2 ^ { x }$ ∙ $8 ^ { x }$ Enter your answer exactly as $2^a$ .

f(x) = $x ^ { 3 }$ $e ^ { - x ^ { 3 } }$ Enter your answer exactly as just $\mathrm { e } ^ { \mathrm { P } } ( \mathrm { x } )$ (Q(x)) where P and Q are polynomials in x in standard form.

2 $e ^ { 3 x + 1 }$ = $e ^ { 2 }$ Enter your answer exactly as x = $\frac { a - \ln b } { c }$ .

Simplify - $9 x$ ∙ $81 ^ { x }$ ∙ $243 ^x$ Enter your answer exactly as $3^ a$ .

$\sqrt { 1 + \ln ( 2 - x ) }$ at x = 1 Enter your answer as just a reduced fraction $\frac { a } { b }$ .