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$f\left(x\right)={e}^{x}+2$
Domain: all real numbers, range: $\left(2,\infty \right),y=2$
$f\left(x\right)=\text{\u2212}{2}^{x}$
$f\left(x\right)={3}^{x+1}$
Domain: all real numbers, range: $\left(0,\infty \right),y=0$
$f\left(x\right)={4}^{x}-1$
$f\left(x\right)=1-{2}^{\text{\u2212}x}$
Domain: all real numbers, range: $\left(\text{\u2212}\infty ,1\right),y=1$
$f\left(x\right)={5}^{x+1}+2$
$f\left(x\right)={e}^{\text{\u2212}x}-1$
Domain: all real numbers, range: $\left(\mathrm{-1},\infty \right),y=\mathrm{-1}$
For the following exercises, write the equation in equivalent exponential form.
${\text{log}}_{3}81=4$
${\text{log}}_{5}1=0$
$\text{log}\phantom{\rule{0.1em}{0ex}}0.1=\mathrm{-1}$
$\text{ln}\left(\frac{1}{{e}^{3}}\right)=\mathrm{-3}$
${e}^{\mathrm{-3}}=\frac{1}{{e}^{3}}$
${\text{log}}_{9}3=0.5$
For the following exercises, write the equation in equivalent logarithmic form.
${2}^{3}=8$
${4}^{\mathrm{-2}}=\frac{1}{16}$
${\text{log}}_{4}\left(\frac{1}{16}\right)=\mathrm{-2}$
${10}^{2}=100$
${\left(\frac{1}{3}\right)}^{3}=\frac{1}{27}$
${e}^{x}=y$
${b}^{3}=45$
${4}^{\mathrm{-3}\text{/}2}=0.125$
${\text{log}}_{4}0.125=-\frac{3}{2}$
For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote.
$f\left(x\right)=3+\text{ln}\phantom{\rule{0.1em}{0ex}}x$
$f\left(x\right)=\text{ln}(x-1)$
Domain: $(1,\infty ),$ range: $\left(\text{\u2212}\infty ,\infty \right),x=1$
$f\left(x\right)=\text{ln}(\text{\u2212}x)$
$f\left(x\right)=1-\text{ln}\phantom{\rule{0.1em}{0ex}}x$
Domain: $\left(0,\infty \right),$ range: $\left(\text{\u2212}\infty ,\infty \right),x=0$
$f\left(x\right)=\text{log}\phantom{\rule{0.1em}{0ex}}x-1$
$f\left(x\right)=\text{ln}(x+1)$
Domain: $\left(\mathrm{-1},\infty \right),$ range: $\left(\text{\u2212}\infty ,\infty \right),x=\mathrm{-1}$
For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms.
$\text{log}{x}^{4}y$
${\text{log}}_{3}\frac{9{a}^{3}}{b}$
$2+3{\text{log}}_{3}a-{\text{log}}_{3}b$
$\text{ln}\phantom{\rule{0.1em}{0ex}}a\sqrt[3]{b}$
${\text{log}}_{5}\sqrt{125x{y}^{3}}$
$\frac{3}{2}+\frac{1}{2}{\text{log}}_{5}x+\frac{3}{2}{\text{log}}_{5}y$
${\text{log}}_{4}\frac{\sqrt[3]{xy}}{64}$
$\text{ln}\left(\frac{6}{\sqrt{{e}^{3}}}\right)$
$-\frac{3}{2}+\text{ln}\phantom{\rule{0.1em}{0ex}}6$
For the following exercises, solve the exponential equation exactly.
${5}^{x}=125$
${e}^{3x}-15=0$
$\frac{\text{ln}\phantom{\rule{0.1em}{0ex}}15}{3}$
${8}^{x}=4$
${3}^{x\text{/}14}=\frac{1}{10}$
$4\xb7{2}^{3x}-20=0$
${7}^{3x-2}=11$
$\frac{2}{3}+\frac{\text{log}\phantom{\rule{0.1em}{0ex}}11}{3\phantom{\rule{0.1em}{0ex}}\text{log}\phantom{\rule{0.1em}{0ex}}7}$
For the following exercises, solve the logarithmic equation exactly, if possible.
${\text{log}}_{3}x=0$
${\text{log}}_{4}\left(x+5\right)=0$
$\text{ln}\sqrt{x+3}=2$
${\text{log}}_{6}\left(x+9\right)+{\text{log}}_{6}x=2$
$x=3$
${\text{log}}_{4}\left(x+2\right)-{\text{log}}_{4}\left(x-1\right)=0$
$\text{ln}\phantom{\rule{0.1em}{0ex}}x+\text{ln}\left(x-2\right)=\text{ln}\phantom{\rule{0.1em}{0ex}}4$
$1+\sqrt{5}$
For the following exercises, use the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.
${\text{log}}_{5}47$
${\text{log}}_{7}82$
$\left(\frac{\text{log}\phantom{\rule{0.1em}{0ex}}82}{\text{log}\phantom{\rule{0.1em}{0ex}}7}\approx 2.2646\right)$
${\text{log}}_{6}103$
${\text{log}}_{0.5}211$
$\left(\frac{\text{log}\phantom{\rule{0.1em}{0ex}}211}{\text{log}\phantom{\rule{0.1em}{0ex}}0.5}\approx -7.7211\right)$
${\text{log}}_{2}\pi $
${\text{log}}_{0.2}0.452$
$\left(\frac{\text{log}\phantom{\rule{0.1em}{0ex}}0.452}{\text{log}\phantom{\rule{0.1em}{0ex}}0.2}\approx 0.4934\right)$
Rewrite the following expressions in terms of exponentials and simplify.
a. $2\phantom{\rule{0.1em}{0ex}}\text{cosh}\left(\text{ln}\phantom{\rule{0.1em}{0ex}}x\right)$ b. $\text{cosh}\phantom{\rule{0.1em}{0ex}}4x+\text{sinh}\phantom{\rule{0.1em}{0ex}}4x$ c. $\text{cosh}\phantom{\rule{0.1em}{0ex}}2x-\text{sinh}\phantom{\rule{0.1em}{0ex}}2x$ d. $\text{ln}\left(\text{cosh}\phantom{\rule{0.1em}{0ex}}x+\text{sinh}\phantom{\rule{0.1em}{0ex}}x\right)+\text{ln}\left(\text{cosh}\phantom{\rule{0.1em}{0ex}}x-\text{sinh}\phantom{\rule{0.1em}{0ex}}x\right)$
[T] The number of bacteria N in a culture after t days can be modeled by the function $N\left(t\right)=1300\xb7{\left(2\right)}^{t\text{/}4}.$ Find the number of bacteria present after 15 days.
$~17,491$
[T] The demand D (in millions of barrels) for oil in an oil-rich country is given by the function $D\left(p\right)=150\xb7{\left(2.7\right)}^{\mathrm{-0.25}p},$ where p is the price (in dollars) of a barrel of oil. Find the amount of oil demanded (to the nearest million barrels) when the price is between $15 and $20.
[T] The amount A of a $100,000 investment paying continuously and compounded for t years is given by $A\left(t\right)=\mathrm{100,000}\xb7{e}^{0.055t}.$ Find the amount A accumulated in 5 years.
Approximately $131,653 is accumulated in 5 years.
[T] An investment is compounded monthly, quarterly, or yearly and is given by the function $A=P{\left(1+\frac{j}{n}\right)}^{nt},$ where $A$ is the value of the investment at time $t,P$ is the initial principle that was invested, $j$ is the annual interest rate, and $n$ is the number of time the interest is compounded per year. Given a yearly interest rate of 3.5% and an initial principle of $100,000, find the amount $A$ accumulated in 5 years for interest that is compounded a. daily, b., monthly, c. quarterly, and d. yearly.
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