지수함수와 로그함수 전체 문제

For the following exercises, solve the equation for $x,$ if there is a solution*.* Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.

$\ln (2x+9)=\ln (-5x)$

For the following exercises, solve the equation for $x,$ if there is a solution*.* Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.

${\log }_{9}(3-x)={\log }_{9}(4x-8)$

For the following exercises, solve the equation for $x,$ if there is a solution*.* Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.

$\log ({x}^{2}+13)=\log (7x+3)$

For the following exercises, solve the equation for $x,$ if there is a solution*.* Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.

$\frac{3}{{\log }_{2}(10)}-\log (x-9)=\log (44)$

For the following exercises, solve the equation for $x,$ if there is a solution*.* Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.

$\ln (x)-\ln (x+3)=\ln (6)$

For the following exercises, solve for the indicated value, and graph the situation showing the solution point.

An account with an initial deposit of $\text{\textdollar}6,500$ earns $7.25\%$ annual interest, compounded continuously. How much will the account be worth after 20 years?

For the following exercises, solve for the indicated value, and graph the situation showing the solution point.

The formula for measuring sound intensity in decibels $D$ is defined by the equation $D=10\log \left(\frac{I}{{I}_{0}}\right),$ where $I$ is the intensity of the sound in watts per square meter and ${I}_{0}={10}^{-12}$ is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of $8.3\cdot {10}^{2}$ watts per square meter?

For the following exercises, solve for the indicated value, and graph the situation showing the solution point.

The population of a small town is modeled by the equation $P=1650{e}^{0.5t}$ where $t$ is measured in years. In approximately how many years will the town’s population reach $20,000?$

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.

$1000{(1.03)}^{t}=5000$ using the common log.

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.

${e}^{5x}=17$ using the natural log

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.

$3{(1.04)}^{3t}=8$ using the common log

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.

${3}^{4x-5}=38$ using the common log

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.

$50{e}^{-0.12t}=10$ using the natural log

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

$7{e}^{3x-5}+7.9=47$

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

$\ln (3)+\ln (4.4x+6.8)=2$

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

$\log (-0.7x-9)=1+5\log (5)$

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

Atmospheric pressure $P$ in pounds per square inch is represented by the formula $P=14.7{e}^{-0.21x},$ where $x$ is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of $8.369$ pounds per square inch? (Hint: there are 5280 feet in a mile)

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

The magnitude M of an earthquake is represented by the equation $M=\frac{2}{3}\log \left(\frac{E}{{E}_{0}}\right)$ where $E$ is the amount of energy released by the earthquake in joules and ${E}_{0}={10}^{4.4}$ is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing $1.4\cdot {10}^{13}$ joules of energy?

Use the definition of a logarithm along with the one-to-one property of logarithms to prove that ${b}^{{\log }_{b}x}=x.$

Recall the formula for continually compounding interest, $y=A{e}^{kt}.$ Use the definition of a logarithm along with properties of logarithms to solve the formula for time $t$ such that $t$ is equal to a single logarithm.