How do you solve the equation $$\text{log}_4(6) - \text{log}_4(4x) = \text{log}_4(63)$$ ?

Combine the logs using the quotient property: $$\text{log}_4igg(\frac{6}{4x}igg) = \text{log}_4(63)$$ . Then set the arguments equal: $$\frac{6}{4x} = 63$$ , leading to $$x = \frac{1}{42}$$ .

How do you solve the equation $$\text{log}_4(7) + \text{log}_4(x - 2) = \text{log}_4(62)$$ ?

Combine the logs using the product property: $$\text{log}_4(7(x - 2)) = \text{log}_4(62)$$ . Set the arguments equal: $$7(x - 2) = 62$$ , leading to $$x = \frac{76}{7}$$ .

How do you solve the equation $$\text{log}_4(6) + \text{log}_4(x) = 2$$ ?

Combine the logs using the product property: $$\text{log}_4(6x) = 2$$ . Set the argument equal to the base raised to the power: $$6x = 4^2$$ , leading to $$x = \frac{8}{3}$$ .

How do you solve the equation $$-8 + \text{log}_6(x + 1) = -7$$ ?

Isolate the log: $$\text{log}_6(x + 1) = 1$$ . Set the argument equal to the base raised to the power: $$x + 1 = 6^1$$ , leading to $$x = 5$$ .

How do you solve the equation $$\text{log}_9(r - 8) = 2$$ ?

Set the argument equal to the base raised to the power: $$r - 8 = 9^2$$ , leading to $$r = 89$$ .

What is the domain of a logarithmic function?

The domain of a logarithmic function $$\text{log}_b(x)$$ is all positive real numbers, i.e., $$x > 0$$ .

What is the range of a logarithmic function?

The range of a logarithmic function $$\text{log}_b(x)$$ is all real numbers, i.e., $$(-\text{∞}, \text{∞})$$ .

What is the relationship between logarithms and exponents?

Logarithms are the inverse operations of exponentiation. If $$b^y = x$$ , then $$\text{log}_b(x) = y$$ .

What is the significance of the base in logarithms?

The base of a logarithm determines the rate of growth of the logarithmic function. Common bases are 10 (common logarithm) and e (natural logarithm).

How can logarithmic equations be used in real-world applications?

Logarithmic equations can model exponential growth or decay, such as population growth, radioactive decay, and sound intensity (decibels).

Solving Log Equations Flashcards

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FLASHCARD QUESTION

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What is the definition of a logarithm?

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