Logarithmic Equations

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What is a logarithmic equation?

A logarithmic equation is an equation that involves a logarithm of an expression set equal to a number or another expression.

How do you solve a basic logarithmic equation, such as \( \log_b(x) = c \)?

To solve \( \log_b(x) = c \), rewrite the equation in exponential form as \( x = b^c \).

What must be true about the base and argument in a logarithmic equation?

The base must be positive and not equal to 1, and the argument must be positive.

How can properties of logarithms simplify solving logarithmic equations?

Properties such as the product, quotient, and power rules can simplify expressions and isolate the variable.

What is the product rule for logarithms?

The product rule states \( \log_b(MN) = \log_b(M) + \log_b(N) \).

What is the power rule for logarithms?

The power rule states \( \log_b(M^n) = n \cdot \log_b(M) \).

What is the quotient rule for logarithms?

The quotient rule states \( \log_b(M/N) = \log_b(M) - \log_b(N) \).

How do you solve \( \log_b(x^2) = \log_b(9) \)?

By the one-to-one property, set \( x^2 = 9 \), solve for \( x \): \( x = 3 \) or \( x = -3 \) (considering restrictions).

Why can extraneous solutions occur in logarithmic equations?

Extraneous solutions occur due to taking roots and combining logs that might not hold under original restrictions.

What is the change of base formula for logarithms?

The change of base formula is \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \), for any positive k.

How does the change of base formula help solve equations?

It allows rewriting logarithms in terms of a different base, often converting them to a base easier to work with, like 10 or e.

How can you verify a solution to a logarithmic equation?

Substitute the solution back into the original equation and check if both sides are equal.

What strategy can be used to solve equations with multiple logarithms?

Use properties of logs to combine into a single logarithm, then rewrite in exponential form to solve.

What role does graphing play in solving logarithmic equations?

Graphing can help visualize solutions and verify answers or identify extraneous solutions.

What is an example of an equation involving a natural logarithm, and how to solve it?

For \( \ln(x) = 2 \), rewrite it as \( x = e^2 \).




Test Your Knowledge

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1. What is a logarithmic equation?

An equation that involves exponential growth.

An equation that measures angles in trigonometric functions.

A logarithmic equation is an equation that involves a logarithm of an expression set equal to a number or another expression.

An equation that solves for the slope of a line.

2. How do you solve a basic logarithmic equation, such as \( \log_b(x) = c \)?

Use the Pythagorean Theorem to find the solution.

To solve \( \log_b(x) = c \), rewrite the equation in exponential form as \( x = b^c \).

Rewrite the equation as \( c = b \cdot x \).

Factor the polynomial to find the solution.

3. What must be true about the base and argument in a logarithmic equation?

The base and argument must both be negative.

The base must be positive and not equal to 1, and the argument must be positive.

Both the base and argument can be zero.

The base and argument must both be even numbers.

4. How can properties of logarithms simplify solving logarithmic equations?

By integrating the functions involved.

Properties such as the product, quotient, and power rules can simplify expressions and isolate the variable.

By converting them to algebraic equations.

By applying statistical methods.

5. What is the product rule for logarithms?

\( \log_b(M/N) = \log_b(M) \cdot \log_b(N) \)

\( \log_b(M+N) = \log_b(M) - \log_b(N) \)

The product rule states \( \log_b(MN) = \log_b(M) + \log_b(N) \).

\( \log_b(M^n) = n + \log_b(M) \)

6. What is the power rule for logarithms?

The power rule states \( \log_b(M^n) = n \cdot \log_b(M) \).

The power rule states \( \log_b(M^n) = n \cdot M \).

The power rule states \( \log_b(M) = n \cdot M \).

The power rule states \( \log_b(M^n) = \frac{1}{n} \cdot \log_b(M) \).

7. What is the quotient rule for logarithms?

The quotient rule states \( \log_b(M/N) = \log_b(M) - \log_b(N) \).

The quotient rule states \( \log_b(M/N) = \log_b(M) + \log_b(N) \).

The quotient rule states \( \log_b(M+N) = \log_b(M) \cdot \log_b(N) \).

The quotient rule states \( \log_b(M \cdot N) = \log_b(M) - \log_b(N) \).

8. How do you solve \( \log_b(x^2) = \log_b(9) \)?

By the one-to-one property, set \( x^2 = 9 \), solve for \( x \): \( x = 3 \) or \( x = -3 \) (considering restrictions).

Use the difference of squares method to solve for x.

Factor \( x^2 \) and recombine terms.

Rewrite as an equation with base e and solve.

9. Why can extraneous solutions occur in logarithmic equations?

Because solutions from squaring logs always add more variability.

Extraneous solutions occur due to taking roots and combining logs that might not hold under original restrictions.

Extraneous solutions occur because logs are not defined for negative numbers.

They simply arise from solving any algebraic equations.

10. What is the change of base formula for logarithms?

\( \log_b(x) = \log(x) - \log(b) \)

\( \log_b(x) = \log(x) - \log(b) \cdot 2 \)

The change of base formula is \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \), for any positive k.

\( \log_b(x) = \log_k(x) + \log_k(b) \)