Logarithms Exercises

Subsection 2.7 Logarithms Exercises

Solve each equation below.
🔗
1. $k(x)=3 e^{2x}+1\text{,}$ solve $k(x)=11$ for $x$
  🔗
  
  🔗
2. $q(x)=-6 \log_3 (x-3)\text{,}$ solve $q(x)=-24$ for $x$
  🔗
  
  🔗
3. $f(x)=2 \log_6 4x\text{,}$ solve $f(x)=4$ for $x$
  🔗
  
  🔗
4. $g(x)=7^{2x-3}-4\text{,}$ solve $g(x)=14$ for $x$
  🔗
  
  🔗
5. $a(x)=5^{3-2x}\text{,}$ $b(x)=5^{-x}\text{,}$ solve $a(x)=b(x)$ for $x$
  🔗
  
  🔗
6. $c(x)=8^{x-1}\text{,}$ $d(x)=2^{x+2}\text{,}$ solve $c(x)=d(x)$ for $x$
  🔗
  
  🔗
🔗
🔗
Solve for $t$ in the equation $A=25\cdot 3^{t/8}\text{.}$
🔗

🔗

🔗
Refer to the population of Albuquerque, NM, for given years as shown in Table 2.7.8.
Table 2.7.8. Population, $P(t)\text{,}$ of Albuquerque $t$ years after 1980
🔗

$t$ 0 10 20 30

$P(t)$ 331,767 386,988 450,557 545,852

🔗
1. Using the population of Albuquerque in 1980 and 2010, build an exponential model for $P(t)\text{.}$
  🔗
  
  🔗
2. Use your exponential model to predict the population of Albuquerque in 1990. How does your prediction compare with the actual population at that time?
  🔗
  
  🔗
3. Use your exponential model to predict in what year the population of Albuquerque will be 700,000. First give your answer in exact form using a logarithm, and then give the decimal approximation.
  🔗
  
  🔗
4. Using the population of Albuquerque in 1980 and 2010, build a linear model for $P(t)\text{.}$
  🔗
  
  🔗
5. Do you think the linear model or the exponential model is a better fit to the data? Explain.
  🔗
  
  🔗
🔗
🔗
The population of Fresno, CA was 430,724 people in 2000. At that time, the population was growing at an annual rate of $1.3\%\text{.}$
🔗
1. Write an exponential equation relating the population, $P(t)\text{,}$ to the number of years, $t\text{,}$ with $t=0$ corresponding to 2000.
  🔗
  
  🔗
2. If the population continued to grow at that rate, what is the population of Fresno in 2015?
  🔗
  
  🔗
3. According to the model, when will the population of Fresno reach 600,000?
  🔗
  
  🔗
🔗
🔗
Shayla opens a banking account with $300. The account earns 2.2% compounded annually. Let $t$ be the number of years the bank account has been open, and $A(t)$ the balance in the account.
🔗
1. Make a table showing the balance, $A(t)\text{,}$ in the account at $t=0\text{,}$ 1, 2, 3, and 4 years.
  🔗
  
  🔗
2. Write an equation for the function $A(t)\text{.}$
  🔗
  
  🔗
3. Use your function equation to determine the balance in 20 years.
  🔗
  
  🔗
4. When will the balance double?
  🔗
  
  🔗
🔗
🔗
Solve. Give your answer in both exact form and as a decimal approximation.
🔗
1. $\displaystyle 6 \cdot 3^{8x} =1000$
  🔗
  
  🔗
2. $\displaystyle 10^{-2v}+365=2118$
  🔗
  
  🔗
🔗
🔗
The intensity of light $d$ meters away from a 100-Watt bulb, measured in Watts per square meter (W/m$^2$), can be modeled by the equation

\begin{equation*} I=a \cdot d^{-2} \end{equation*}

🔗
1. If the intensity of light 1.5 meters from the bulb is 3.53 W/m$^2\text{,}$ find the constant $a$ and write the equation to model the intensity of light $d$ meters away from a 100-Watt bulb.
  🔗
  
  🔗
2. What will the intensity of light be 2.5 meters away from the bulb?
  🔗
  
  🔗
3. At what distance away from the bulb will the light have an intensity of 4 W/m$^2\text{.}$
  🔗
  
  🔗
4. Assuming that both distance and light intensity are positive, write an equation for the inverse function that gives the distance from a 100-Watt bulb as a function of the light intensity measured at that distance.
  🔗
  
  🔗
🔗
🔗
Simplify each expression.
🔗
1. $\displaystyle 2\sqrt[3]{x^2}(x^4-5x^3-x)$
  🔗
  
  🔗
2. $\displaystyle \frac{x^{-2}-x+3x^2}{x^{-2}}$
  🔗
  
  🔗
🔗
🔗

\(t\)	0	10	20	30
\(P(t)\)	331,767	386,988	450,557	545,852

🔗

Prev Top Next