지수함수와 로그함수 전체 문제

Use the definition of a logarithm to solve. $-5{\log }_{7}(10n)=5.$

Use the definition of a logarithm to find the exact solution for $9+6\ln (a+3)=33.$

Use the one-to-one property of logarithms to find an exact solution for ${\log }_{8}(7)+{\log }_{8}(-4x)={\log }_{8}(5).$ If there is no solution, write no solution.

Use the one-to-one property of logarithms to find an exact solution for $\ln (5)+\ln (5{x}^{2}-5)=\ln (56).$ If there is no solution, write no solution.

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 large orchestra with a sound intensity of $6.3\cdot {10}^{-3}$ watts per square meter?

The population of a city is modeled by the equation $P(t)=256,114{e}^{0.25t}$ where $t$ is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million?

Find the inverse function ${f}^{-1}$ for the exponential function $f(x)=2\cdot {e}^{x+1}-5.$

Find the inverse function ${f}^{-1}$ for the logarithmic function $f(x)=0.25\cdot {\log }_{2}({x}^{3}+1).$

For the following exercises, use this scenario: A doctor prescribes $300$ milligrams of a therapeutic drug that decays by about $17\%$ each hour.

To the nearest minute, what is the half-life of the drug?

For the following exercises, use this scenario: A doctor prescribes $300$ milligrams of a therapeutic drug that decays by about $17\%$ each hour.

Write an exponential model representing the amount of the drug remaining in the patient’s system after $t$ hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after $24$ hours. Round to the nearest hundredth of a gram.

For the following exercises, use this scenario: A soup with an internal temperature of $350^\circ$ Fahrenheit was taken off the stove to cool in a $\text{71°F}$ room. After fifteen minutes, the internal temperature of the soup was $\text{175°F}.$

Use Newton’s Law of Cooling to write a formula that models this situation.

For the following exercises, use this scenario: A soup with an internal temperature of $350^\circ$ Fahrenheit was taken off the stove to cool in a $\text{71°F}$ room. After fifteen minutes, the internal temperature of the soup was $\text{175°F}.$

How many minutes will it take the soup to cool to $\text{85°F?}$

For the following exercises, use this scenario: The equation $N(t)=\frac{1200}{1+199{e}^{-0.625t}}$ models the number of people in a school who have heard a rumor after $t$ days.

How many people started the rumor?

For the following exercises, use this scenario: The equation $N(t)=\frac{1200}{1+199{e}^{-0.625t}}$ models the number of people in a school who have heard a rumor after $t$ days.

To the nearest tenth, how many days will it be before the rumor spreads to half the carrying capacity?

For the following exercises, use this scenario: The equation $N(t)=\frac{1200}{1+199{e}^{-0.625t}}$ models the number of people in a school who have heard a rumor after $t$ days.

What is the carrying capacity?

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.

Find a formula for an exponential equation that goes through the points $(-2,100)$ and $(0,4).$ Then express the formula as an equivalent equation with base e.

What is the carrying capacity for a population modeled by the logistic equation $P(t)=\frac{250,000}{1+499{e}^{-0.45t}}?$ What is the initial population for the model?

The population of a culture of bacteria is modeled by the logistic equation $\,P(t)=\frac{14,250}{1+29{e}^{-0.62t}},$ where $t$ is in days. To the nearest tenth, how many days will it take the culture to reach $75\%$ of its carrying capacity?