Step-by-Step Guidance for Solving Exponential and Logarithmic Equations

Q1. Solve the exponential equation: $3^{x-2} = 27$

Background

Topic: Exponential Equations

This question tests your ability to solve exponential equations by expressing both sides with the same base and using properties of exponents.

Key Terms and Formulas

• Exponential Equation: An equation where the variable appears in the exponent.

• Base-Exponent Property: $a^x = a^y \iff x = y$ (if $a > 0$ and $a \neq 1$)

Step-by-Step Guidance

1. Recognize that $27$can be written as a power of $3$. What exponent of $3$gives $27$?

2. Rewrite the equation so both sides have the same base: $3^{x-2} = 3^{?}$

3. Apply the property: If $a^m = a^n$, then $m = n$.

4. Set the exponents equal to each other and solve for $x$.

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Q2. Solve the exponential equation: $4^x = 64$

Background

Topic: Exponential Equations

This question tests your ability to solve for the exponent when both sides can be written as powers of the same base.

Key Terms and Formulas

• Exponential Equation: An equation where the variable is in the exponent.

• Base-Exponent Property: $a^x = a^y \iff x = y$

Step-by-Step Guidance

1. Express $64$as a power of $4$.

2. Rewrite the equation so both sides have the same base: $4^x = 4^{?}$

3. Set the exponents equal to each other and solve for $x$.

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Q3. Solve the exponential equation: $15e^{4x+2} = 30$

Background

Topic: Exponential Equations with $e$

This question tests your ability to solve for $x$ when the variable is in the exponent and the base is $e$ (Euler's number).

Key Terms and Formulas

• Natural Exponential Function: $e^x$

• Natural Logarithm: $\ln(x)$, the inverse of $e^x$

• Solving Steps: Isolate the exponential term, then take the natural logarithm of both sides.

Step-by-Step Guidance

1. Divide both sides by $15$to isolate$e^{4x+2}$.

2. Take the natural logarithm ($\ln$) of both sides to bring down the exponent.

3. Use the property $\ln(e^y) = y$ to simplify.

4. Solve for $x$ by isolating it on one side of the equation.

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Q4. Solve the exponential equation: $200 = 2^t$

Background

Topic: Exponential Equations

This question tests your ability to solve for the exponent when the base is $2$ and the variable is in the exponent.

Key Terms and Formulas

• Exponential Equation: $a^x = b$

• Logarithm: $x = \log_a b$

• Change of Base Formula: $\log_a b = \frac{\ln b}{\ln a}$

Step-by-Step Guidance

1. Rewrite the equation as $2^t = 200$ if needed.

2. Take the logarithm of both sides (you can use natural log or log base 2).

3. Use the property $\log_a a^x = x$ to bring down the exponent.

4. Solve for $t$ by isolating it.

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Q5. Solve the exponential equation: $5^{x-4} = 4$

Background

Topic: Exponential Equations

This question tests your ability to solve for the exponent when the base and exponent are both variables or constants.

Key Terms and Formulas

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

• Logarithm: $x = \log_a d$

Step-by-Step Guidance

1. Take the logarithm of both sides (any base, but natural log is common).

2. Use the property $\log_a a^x = x$ to bring down the exponent.

3. Solve for $x$ by isolating it.

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Q6. Solve the logarithmic equation: $\log_2 3x = -2$

Background

Topic: Logarithmic Equations

This question tests your ability to convert a logarithmic equation to its equivalent exponential form and solve for $x$.

Key Terms and Formulas

• Logarithmic Equation: $\log_a b = c \iff b = a^c$

Step-by-Step Guidance

1. Rewrite the logarithmic equation in exponential form: $3x = 2^{-2}$

2. Solve for $x$ by dividing both sides by $3$.

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Q7. Solve the logarithmic equation: $\log_2 (x+1) - \log_2 (x-1) = 3$

Background

Topic: Logarithmic Equations

This question tests your ability to use properties of logarithms to combine and solve equations.

Key Terms and Formulas

• Logarithm Subtraction Property: $\log_a M - \log_a N = \log_a \left(\frac{M}{N}\right)$

• Exponential Form: $\log_a b = c \iff b = a^c$

Step-by-Step Guidance

1. Combine the logarithms using the subtraction property: $\log_2 \left(\frac{x+1}{x-1}\right) = 3$

2. Rewrite the equation in exponential form: $\frac{x+1}{x-1} = 2^3$

3. Solve for $x$ by cross-multiplying and isolating $x$.

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Q8. Solve the logarithmic equation: $\log_3 (x+14) - \log_3 (x+6) + \log_3 x = 2$

Background

Topic: Logarithmic Equations

This question tests your ability to use properties of logarithms to combine multiple terms and solve for $x$.

Key Terms and Formulas

• Logarithm Properties: $\log_a M + \log_a N = \log_a (MN)$ and $\log_a M - \log_a N = \log_a \left(\frac{M}{N}\right)$

• Exponential Form: $\log_a b = c \iff b = a^c$

Step-by-Step Guidance

1. Combine the logarithms into a single logarithm using the properties above.

2. Rewrite the equation in exponential form.

3. Solve for $x$ by simplifying the resulting equation.

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Q9. Solve the logarithmic equation: $2\log_5 (x+4) = 1$

Background

Topic: Logarithmic Equations

This question tests your ability to use the power property of logarithms and solve for $x$.

Key Terms and Formulas

• Power Property: $k\log_a b = \log_a b^k$

• Exponential Form: $\log_a b = c \iff b = a^c$

Step-by-Step Guidance

1. Use the power property to rewrite the left side: $2\log_5 (x+4) = \log_5 (x+4)^2$

2. Set $\log_5 (x+4)^2 = 1$ and rewrite in exponential form.

3. Solve for $x$ by taking the square root and isolating $x$.

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Q10. Solve the logarithmic equation: $\ln(x+4) = 4$

Background

Topic: Logarithmic Equations (Natural Logarithm)

This question tests your ability to solve for $x$ when given a natural logarithm equation.

Key Terms and Formulas

• Natural Logarithm: $\ln(x)$ is the logarithm base $e$.

• Exponential Form: $\ln a = b \iff a = e^b$

Step-by-Step Guidance

1. Rewrite the equation in exponential form: $x+4 = e^4$

2. Solve for $x$ by subtracting $4$ from both sides.

Try solving on your own before revealing the answer!

Q11. Solve the logarithmic equation: $3\log_{10} x + \log_{10} 4 = 14$

Background

Topic: Logarithmic Equations (Common Logarithm)

This question tests your ability to use properties of logarithms to combine terms and solve for $x$.

Key Terms and Formulas

• Power Property: $k\log_a b = \log_a b^k$

• Sum Property: $\log_a M + \log_a N = \log_a (MN)$

• Exponential Form: $\log_a b = c \iff b = a^c$

Step-by-Step Guidance

1. Use the power property to rewrite $3\log_{10} x$ as $\log_{10} x^3$.

2. Combine the logarithms: $\log_{10} x^3 + \log_{10} 4 = \log_{10} (4x^3)$

3. Set $\log_{10} (4x^3) = 14$ and rewrite in exponential form.

4. Solve for $x$ by isolating $x^3$ and then taking the cube root.

Try solving on your own before revealing the answer!