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 how to solve equations involving natural logarithms.

Key Terms and Formulas

• Natural logarithm: $\ln a$ is the logarithm base $e$ of $a$.

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

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

Step-by-Step Guidance

1. Combine the two logarithms using the property $\ln a + \ln b = \ln(ab)$.

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

3. Exponentiate both sides to eliminate the logarithm: $x(x-3) = 1$.

4. Rewrite the equation as a quadratic and set it equal to zero.

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

Background

Topic: Exponential and Logarithmic Equations

This question tests your ability to manipulate logarithmic equations and solve for the variable.

Key Terms and Formulas

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

• If $\ln y = \ln k$, then $y = k$ (as long as $y, k > 0$).

Step-by-Step Guidance

1. Combine the left side using $\ln a + \ln b = \ln(ab)$ to get $\ln(x(x-3))$.

2. Set $\ln(x(x-3)) = \ln 4$.

3. Since the logarithms are equal, set their arguments equal: $x(x-3) = 4$.

4. Rewrite as a quadratic equation and prepare to solve for $x$.

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

Background

Topic: Exponential Equations

This question tests your ability to solve equations where the variable is in the exponent.

Key Terms and Formulas

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

• Take logarithms of both sides to bring down the exponent.

• Use $\log_a b = \frac{\ln b}{\ln a}$ if needed.

Step-by-Step Guidance

1. Rewrite $21$ as $7^1 \times 3$ or just 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 bring down the exponent.

4. Isolate $x$ by solving the resulting linear equation.

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

Background

Topic: Exponential Equations

This question tests your ability to solve for $x$ when it appears in the exponent.

Key Terms and Formulas

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

• Take logarithms of both sides to solve for $x$.

Step-by-Step Guidance

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

2. Use the property $\ln(a^b) = b\ln a$ to bring down the exponent.

3. Expand and isolate $x$ in the resulting linear equation.

4. Prepare to solve for $x$ by moving terms appropriately.

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

Background

Topic: Logarithmic Equations

This question tests your ability to solve for $x$ in a logarithmic equation.

Key Terms and Formulas

• Property: $a\ln x = \ln x^a$

• To solve $\ln y = k$, exponentiate both sides: $y = e^k$

Step-by-Step Guidance

1. Divide both sides by 3 to isolate $\ln x$.

2. Exponentiate both sides to solve for $x$.

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

Background

Topic: Exponential Equations

This question tests your understanding of the properties of the exponential function and how to solve for the exponent when the output is 1.

Key Terms and Formulas

• Property: $e^k = 1$ if and only if $k = 0$

Step-by-Step Guidance

1. Recall that $e^y = 1$ only when $y = 0$.

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

3. Rewrite as a quadratic equation and prepare to solve for $x$.

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

Background

Topic: Logarithmic Properties

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

Key Terms and Formulas

• Property: $\log_b a + \log_b c = \log_b (ac)$

• Evaluate $\log_b d$ by expressing $d$ as a power of $b$ if possible.

Step-by-Step Guidance

1. Combine the two logarithms using $\log_b a + \log_b c = \log_b (ac)$.

2. Multiply $9 \times 25$ to get the argument of the single logarithm.

3. Express the result as $\log_{15} (225)$ and see if $225$ can be written as a power of $15$.

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

Background

Topic: Exponential and Logarithmic Properties

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

Key Terms and Formulas

• $e^{a\ln b} = b^a$

• $e^{A+B} = e^A \cdot e^B$

Step-by-Step Guidance

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

2. Combine the exponents: $e^{\ln 5^3 + \ln 2^7} = e^{\ln(5^3 \cdot 2^7)}$.

3. Recall that $e^{\ln k} = k$ for $k > 0$.

4. Multiply $5^3$ and $2^7$ to get the final value.

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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 with given values.

Key Terms and Formulas

• $\log_b \frac{A}{B} = \log_b A - \log_b B$

• $\log_b A^k = k \log_b A$

• $\sqrt{x} = x^{1/2}$, $\sqrt[3]{z} = z^{1/3}$

Step-by-Step Guidance

1. Rewrite $\sqrt{x}$ as $x^{1/2}$ and $\sqrt[3]{z}$ as $z^{1/3}$.

2. Apply the logarithm power rule: $\log_b x^{1/2} = \frac{1}{2} \log_b x$ and $\log_b z^{1/3} = \frac{1}{3} \log_b z$.

3. Use the quotient rule: $\log_b \frac{\sqrt{x}}{\sqrt[3]{z}} = \frac{1}{2} \log_b x - \frac{1}{3} \log_b z$.

4. Substitute the given values for $\log_b x$ and $\log_b z$.

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