Q1. Solve the equation: $\ln x + \ln(x - 3) = 0$

Background

Topic: Exponential and Logarithmic Equations

This question tests your understanding of logarithmic properties and solving equations involving logarithms.

Key Terms and Formulas:

• $\ln a + \ln b = \ln(ab)$ (Logarithm addition property)

• To solve $\ln y = 0$, recall that $y = 1$.

Step-by-Step Guidance

1. Combine the logarithms using the property: $\ln x + \ln(x - 3) = \ln[x(x - 3)]$.

2. Set the combined logarithm equal to zero: $\ln[x(x - 3)] = 0$.

3. Recall that $\ln y = 0$ means $y = 1$. Set $x(x - 3) = 1$.

4. Write the resulting quadratic equation: $x^2 - 3x - 1 = 0$.

Try solving on your own before revealing the answer!

Q2. Solve the equation: $\ln x + \ln(x - 3) = \ln 4$

Background

Topic: Logarithmic Equations

This question tests your ability to manipulate logarithmic expressions and solve for $x$.

Key Terms and Formulas:

• $\ln a + \ln b = \ln(ab)$

• If $\ln y = \ln k$, then $y = k$.

Step-by-Step Guidance

1. Combine the logarithms: $\ln x + \ln(x - 3) = \ln[x(x - 3)]$.

2. Set equal to $\ln 4$: $\ln[x(x - 3)] = \ln 4$.

3. Since $\ln y = \ln k$ implies $y = k$, set $x(x - 3) = 4$.

4. Write the quadratic equation: $x^2 - 3x - 4 = 0$.

Try solving on your own before revealing the answer!

Q3. Solve the equation: $7^{2x+1} = 21$

Background

Topic: Exponential Equations

This question tests your ability to solve equations involving exponents by using logarithms.

Key Terms and Formulas:

• Exponential equation: $a^{bx+c} = d$

• Take logarithms to solve for $x$.

Step-by-Step Guidance

1. Rewrite $21$ as $7^1 \times 3$ to see if bases can be matched, or take the natural logarithm of both sides.

2. Apply $\ln$ to both sides: $\ln(7^{2x+1}) = \ln(21)$.

3. Use the property $\ln(a^b) = b \ln a$ to get $(2x+1)\ln 7 = \ln 21$.

4. Isolate $x$ by rearranging: $2x+1 = \frac{\ln 21}{\ln 7}$.

Try solving on your own before revealing the answer!

Q4. Solve the equation: $3^{5x+2} = 12$

Background

Topic: Exponential Equations

This question tests your ability to solve for $x$ in an exponential equation using logarithms.

Key Terms and Formulas:

• Exponential equation: $a^{bx+c} = d$

• Take logarithms to solve for $x$.

Step-by-Step Guidance

1. Take the natural logarithm of both sides: $\ln(3^{5x+2}) = \ln(12)$.

2. Apply $\ln(a^b) = b \ln a$ to get $(5x+2)\ln 3 = \ln 12$.

3. Isolate $x$: $5x+2 = \frac{\ln 12}{\ln 3}$.

4. Solve for $x$ by rearranging: $x = \frac{1}{5}\left(\frac{\ln 12}{\ln 3} - 2\right)$.

Try solving on your own before revealing the answer!

Q5. Solve the equation: $3\ln x = 5$

Background

Topic: Logarithmic Equations

This question tests your ability to solve for $x$ using properties of logarithms and exponentials.

Key Terms and Formulas:

• $\ln x$ is the natural logarithm of $x$.

• Exponentiate both sides to solve for $x$.

Step-by-Step Guidance

1. Divide both sides by 3: $\ln x = \frac{5}{3}$.

2. Exponentiate both sides: $x = e^{\frac{5}{3}}$.

Try solving on your own before revealing the answer!

Q6. Solve the equation: $e^{2x^2 - 7x - 15} = 1$

Background

Topic: Exponential Equations

This question tests your ability to solve for $x$ when the exponent of $e$ equals zero.

Key Terms and Formulas:

• $e^y = 1$ if and only if $y = 0$.

Step-by-Step Guidance

1. Set the exponent equal to zero: $2x^2 - 7x - 15 = 0$.

2. This is a quadratic equation; use the quadratic formula to solve for $x$.

3. Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 2$, $b = -7$, $c = -15$.

Try solving on your own before revealing the answer!

Q7. Find the exact value: $\log_{15} 9 + \log_{15} 25$

Background

Topic: Logarithmic Properties

This question tests your ability to use logarithm addition properties and evaluate logarithms.

Key Terms and Formulas:

• $\log_a b + \log_a c = \log_a (bc)$

Step-by-Step Guidance

1. Combine the logs: $\log_{15} 9 + \log_{15} 25 = \log_{15} (9 \times 25)$.

2. Calculate $9 \times 25 = 225$.

3. Express $225$ as a power of $15$ if possible.

Try solving on your own before revealing the answer!

Q8. Find the exact value: $e^{3 \ln 5 + 7 \ln 2}$

Background

Topic: Exponential and Logarithmic Properties

This question tests your ability to simplify expressions using properties of logarithms and exponents.

Key Terms and Formulas:

• $e^{a \ln b} = b^a$

• $e^{\ln b + \ln c} = bc$

Step-by-Step Guidance

1. Rewrite $3 \ln 5$ as $\ln 5^3$ and $7 \ln 2$ as $\ln 2^7$.

2. Add the logarithms: $\ln 5^3 + \ln 2^7 = \ln(5^3 \times 2^7)$.

3. Exponentiate: $e^{\ln(5^3 \times 2^7)} = 5^3 \times 2^7$.

Try solving on your own before revealing the answer!

Q9. Find the exact value: $\log_b \frac{\sqrt{x}}{\sqrt[3]{z}}$ when $\log_b x = 2.4$ and $\log_b z = 3.6$

Background

Topic: Logarithmic Properties

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

Key Terms and Formulas:

• $\log_b \sqrt{x} = \frac{1}{2} \log_b x$

• $\log_b \sqrt[3]{z} = \frac{1}{3} \log_b z$

• $\log_b \frac{a}{b} = \log_b a - \log_b b$

Step-by-Step Guidance

1. Apply the root properties: $\log_b \sqrt{x} = \frac{1}{2} \log_b x$, $\log_b \sqrt[3]{z} = \frac{1}{3} \log_b z$.

2. Apply the quotient property: $\log_b \frac{\sqrt{x}}{\sqrt[3]{z}} = \log_b \sqrt{x} - \log_b \sqrt[3]{z}$.

3. Substitute the given values: $\frac{1}{2} \times 2.4 - \frac{1}{3} \times 3.6$.

Try solving on your own before revealing the answer!