Logarithmic Expressions and Applications – Precalculus Study Guidance

Q1. Express log6 216 = 3 in exponential form.

Background

Topic: Logarithmic and Exponential Relationships

This question tests your understanding of how to convert a logarithmic equation to its equivalent exponential form.

Key Terms and Formulas

• Logarithmic form: $\log_b a = c$

• Exponential form: $b^c = a$

Step-by-Step Guidance

1. Identify the base ($b$), the argument ($a$), and the result ($c$) in the logarithmic equation.

2. Recall that $\log_b a = c$ means $b$ raised to the power $c$ equals $a$.

3. Write the equivalent exponential equation using the identified values.

Try solving on your own before revealing the answer!

Final Answer: $6^3 = 216$

We converted the logarithmic equation to exponential form by raising the base to the exponent.

Q2. Express log3 81 = 4 in exponential form.

Background

Topic: Logarithmic and Exponential Relationships

This question tests your ability to convert logarithmic equations to exponential equations.

Key Terms and Formulas

• Logarithmic form: $\log_b a = c$

• Exponential form: $b^c = a$

Step-by-Step Guidance

1. Identify the base ($b$), argument ($a$), and exponent ($c$) in the logarithmic equation.

2. Recall the conversion: $\log_b a = c$ is equivalent to $b^c = a$.

3. Write the exponential equation using these values.

Try solving on your own before revealing the answer!

Final Answer: $3^4 = 81$

The logarithmic equation was rewritten in exponential form.

Q3. Express log3 (1/9) = −2 in exponential form.

Background

Topic: Logarithmic and Exponential Relationships

This question tests your ability to convert logarithmic equations with negative exponents to exponential form.

Key Terms and Formulas

• Logarithmic form: $\log_b a = c$

• Exponential form: $b^c = a$

Step-by-Step Guidance

1. Identify the base ($b$), argument ($a$), and exponent ($c$).

2. Recall that a negative exponent means the reciprocal: $b^{-2} = \frac{1}{b^2}$.

3. Write the exponential equation using these values.

Try solving on your own before revealing the answer!

Final Answer: $3^{-2} = \frac{1}{9}$

Negative exponents indicate reciprocals in exponential form.

Q4. Express log64 8 = 1/2 in exponential form.

Background

Topic: Logarithmic and Exponential Relationships

This question tests your understanding of fractional exponents in logarithmic and exponential forms.

Key Terms and Formulas

• Logarithmic form: $\log_b a = c$

• Exponential form: $b^c = a$

• Fractional exponents: $b^{1/2}$ means the square root of $b$.

Step-by-Step Guidance

1. Identify the base ($b$), argument ($a$), and exponent ($c$).

2. Recall that $b^{1/2}$ is the square root of $b$.

3. Write the exponential equation using these values.

Try solving on your own before revealing the answer!

Final Answer: $64^{1/2} = 8$

Fractional exponents represent roots in exponential form.

Q5. Express log7 = 0.845 in exponential form.

Background

Topic: Logarithmic and Exponential Relationships

This question tests your ability to convert logarithmic equations with decimal exponents to exponential form.

Key Terms and Formulas

• Logarithmic form: $\log_b a = c$

• Exponential form: $b^c = a$

Step-by-Step Guidance

1. Identify the base ($b$), argument ($a$), and exponent ($c$).

2. Recall the conversion: $\log_b a = c$ is equivalent to $b^c = a$.

3. Write the exponential equation using these values.

Try solving on your own before revealing the answer!

Final Answer: $7^{0.845} = a$

The logarithmic equation was rewritten in exponential form.

Q6. Express log1/2 32 = −5 in exponential form.

Background

Topic: Logarithmic and Exponential Relationships

This question tests your ability to convert logarithmic equations with fractional bases and negative exponents to exponential form.

Key Terms and Formulas

• Logarithmic form: $\log_b a = c$

• Exponential form: $b^c = a$

• Negative exponents: $b^{-5} = \frac{1}{b^5}$

Step-by-Step Guidance

1. Identify the base ($b$), argument ($a$), and exponent ($c$).

2. Recall that a negative exponent means the reciprocal.

3. Write the exponential equation using these values.

Try solving on your own before revealing the answer!

Final Answer: $(\frac{1}{2})^{-5} = 32$

Negative exponents indicate reciprocals in exponential form.