College Algebra: Exponential, Logarithmic, and Logistic Functions Review Guidance

Q1. Expand using properties of exponents: $\log_3 \left( \frac{x^3 y}{3z} \right)$

Background

Topic: Properties of Logarithms

This question tests your ability to use the properties of logarithms to expand a logarithmic expression involving products, quotients, and exponents.

Key Terms and Formulas:

• Product Rule: $\log_b(MN) = \log_b M + \log_b N$

• Quotient Rule: $\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$

• Power Rule: $\log_b(M^k) = k \log_b M$

Step-by-Step Guidance

1. First, recognize that the argument inside the logarithm is a quotient: $\frac{x^3 y}{3z}$.

2. Apply the quotient rule to separate the numerator and denominator:

   $\log_3 \left( \frac{x^3 y}{3z} \right) = \log_3(x^3 y) - \log_3(3z)$

3. Next, apply the product rule to both the numerator and denominator where appropriate:

   For the numerator: $\log_3(x^3 y) = \log_3(x^3) + \log_3(y)$

   For the denominator: $\log_3(3z) = \log_3(3) + \log_3(z)$

4. Now, use the power rule to simplify $\log_3(x^3)$:

   $\log_3(x^3) = 3 \log_3(x)$

Try solving on your own before revealing the answer!

Q2. Expand using properties of exponents: $\log_8 \sqrt[3]{x x}$

Background

Topic: Properties of Logarithms and Radicals

This question tests your ability to rewrite radicals as exponents and then expand logarithmic expressions.

Key Terms and Formulas:

• Radical to Exponent: $\sqrt[n]{a^m} = a^{m/n}$

• Product Rule: $\log_b(MN) = \log_b M + \log_b N$

• Power Rule: $\log_b(M^k) = k \log_b M$

Step-by-Step Guidance

1. First, rewrite the radical $\sqrt[3]{x x}$ as an exponent. Remember that $x x = x^2$.

2. So, $\sqrt[3]{x^2} = x^{2/3}$.

3. Now, apply the power rule to the logarithm:

   $\log_8(x^{2/3}) = \frac{2}{3} \log_8(x)$

Try solving on your own before revealing the answer!

Q3. Solve: $3 \log_2 x + 1 = 7$

Background

Topic: Solving Logarithmic Equations

This question tests your ability to solve for $x$ in an equation involving logarithms and constants.

Key Terms and Formulas:

• Logarithmic Equation: An equation that involves a logarithm with a variable inside.

• Isolate the logarithm before converting to exponential form.

• Exponentiate both sides to solve for the variable.

Step-by-Step Guidance

1. Subtract 1 from both sides to isolate the logarithmic term:

   $3 \log_2 x = 6$

2. Divide both sides by 3 to further isolate $\log_2 x$:

   $\log_2 x = 2$

3. Rewrite the logarithmic equation in exponential form:

   $x = 2^2$

Try solving on your own before revealing the answer!

Q4. Solve: $\log_{\sqrt{10}} 3 = x$

Background

Topic: Logarithmic and Exponential Equations

This question tests your understanding of converting between logarithmic and exponential forms.

Key Terms and Formulas:

• Logarithmic to Exponential Form: $\log_b a = c \iff b^c = a$

Step-by-Step Guidance

1. Rewrite the logarithmic equation in exponential form:

   $\left( \sqrt{10} \right)^x = 3$

2. Recall that $\sqrt{10} = 10^{1/2}$, so substitute this in:

   $(10^{1/2})^x = 3$

3. Use the property of exponents: $(a^m)^n = a^{mn}$, so:

   $10^{x/2} = 3$

Try solving on your own before revealing the answer!

Q5. Simplify: $\ln \left( \frac{1}{e^7} \right)$

Background

Topic: Properties of Natural Logarithms

This question tests your ability to use properties of logarithms, especially with the natural logarithm and exponents.

Key Terms and Formulas:

• Quotient Rule: $\ln \left( \frac{1}{a} \right) = -\ln a$

• Power Rule: $\ln(a^k) = k \ln a$

• Natural Logarithm: $\ln e = 1$

Step-by-Step Guidance

1. Apply the quotient rule to rewrite the expression:

   $\ln \left( \frac{1}{e^7} \right) = -\ln(e^7)$

2. Apply the power rule to $\ln(e^7)$:

   $\ln(e^7) = 7 \ln e$

3. Recall that $\ln e = 1$.

Try solving on your own before revealing the answer!

Q6. Earthquake Magnitude Comparison: The first earthquake had $R=6.1$, the second $R=6.9$. Use $\log \left( \frac{a}{T} \right) + B$ to find how many times more powerful the second quake was (assume $B$ and $T$ are equal for both).

Background

Topic: Logarithmic Scales and Earthquake Magnitude

This question tests your understanding of how logarithmic scales (like the Richter scale) compare magnitudes and how to interpret the difference in terms of power.

Key Terms and Formulas:

• Richter Scale: $R = \log \left( \frac{a}{T} \right) + B$

• To compare two earthquakes, set up the difference: $R_2 - R_1 = \log \left( \frac{a_2}{T} \right) - \log \left( \frac{a_1}{T} \right)$

• Use the property: $\log A - \log B = \log \left( \frac{A}{B} \right)$

Step-by-Step Guidance

1. Write the Richter scale equation for both earthquakes:

   $R_1 = \log \left( \frac{a_1}{T} \right) + B$

   $R_2 = \log \left( \frac{a_2}{T} \right) + B$

2. Subtract the first equation from the second to compare their powers:

   $R_2 - R_1 = \log \left( \frac{a_2}{T} \right) - \log \left( \frac{a_1}{T} \right)$

3. Use the logarithm property to combine:

   $R_2 - R_1 = \log \left( \frac{a_2}{a_1} \right)$

4. Exponentiate both sides to solve for $\frac{a_2}{a_1}$:

   $10^{R_2 - R_1} = \frac{a_2}{a_1}$

Try solving on your own before revealing the answer!

Q7. Write the exponential function for an initial population of 67,000, decreasing at 1.67% per year.

Background

Topic: Exponential Decay Functions

This question tests your ability to write an exponential decay function given an initial value and a percent decrease per time period.

Key Terms and Formulas:

• Exponential Decay: $P(t) = P_0 (1 - r)^t$

• $P_0$ = initial population, $r$ = rate of decrease (as a decimal), $t$ = time in years

Step-by-Step Guidance

1. Identify the initial population: $P_0 = 67,000$.

2. Convert the percent decrease to a decimal: $1.67\% = 0.0167$.

3. Plug these values into the exponential decay formula:

   $P(t) = 67,000 \times (1 - 0.0167)^t$

Try writing the function before revealing the answer!

Q8. Write the exponential function for an initial population of 67,000, increasing at 1.67% per year.

Background

Topic: Exponential Growth Functions

This question tests your ability to write an exponential growth function given an initial value and a percent increase per time period.

Key Terms and Formulas:

• Exponential Growth: $P(t) = P_0 (1 + r)^t$

• $P_0$ = initial population, $r$ = rate of increase (as a decimal), $t$ = time in years

Step-by-Step Guidance

1. Identify the initial population: $P_0 = 67,000$.

2. Convert the percent increase to a decimal: $1.67\% = 0.0167$.

3. Plug these values into the exponential growth formula:

   $P(t) = 67,000 \times (1 + 0.0167)^t$

Try writing the function before revealing the answer!

Q9. Rewrite as an exponential expression: $\log_3 8 = x$

Background

Topic: Logarithmic and Exponential Equations

This question tests your ability to convert a logarithmic equation to its equivalent exponential form.

Key Terms and Formulas:

• Logarithmic to Exponential Form: $\log_b a = c \iff b^c = a$

Step-by-Step Guidance

1. Identify the base ($3$), the argument ($8$), and the result ($x$).

2. Rewrite the equation in exponential form:

   $3^x = 8$

Try writing the exponential form before revealing the answer!

Q10. Condense and simplify: $2 \ln(xy^2) - 3 \ln(y)$

Background

Topic: Properties of Logarithms

This question tests your ability to use the properties of logarithms to combine and simplify expressions.

Key Terms and Formulas:

• Power Rule: $k \ln a = \ln(a^k)$

• Product Rule: $\ln(ab) = \ln a + \ln b$

• Quotient Rule: $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$

Step-by-Step Guidance

1. Apply the power rule to both terms:

   $2 \ln(xy^2) = \ln((xy^2)^2) = \ln(x^2 y^4)$

   $3 \ln(y) = \ln(y^3)$

2. Combine using the quotient rule:

   $\ln(x^2 y^4) - \ln(y^3) = \ln\left(\frac{x^2 y^4}{y^3}\right)$

3. Simplify the expression inside the logarithm:

   $\frac{x^2 y^4}{y^3} = x^2 y^{4-3} = x^2 y$

Try condensing and simplifying before revealing the answer!