Standard 4.1 - 4.4 Review

Front

What is the exponential form of the logarithmic equation \( \log_3(x+2) = 4 \)?

Back

The exponential form is \( 3^4 = x + 2 \).

Front

How do you solve the equation \( \log_x(9) = 2 \)?

Back

To solve, rewrite it in exponential form: \( x^2 = 9 \), which gives \( x = 3 \).

Front

What is the result of condensing the equation \( \ln(4x-5) + \ln(2) = \ln(2x+11) \)?

Back

The condensed form is \( \ln(8x-10) = \ln(2x+11) \).

Front

True or False: Extraneous solutions occur when the argument of a logarithm is less than or equal to 0 or when the base is less than or equal to 0 or equals 1.

Back

True.

Front

How do you solve the equation \( \log(x+2) + 2\log(5) = 2 \)?

Back

First, rewrite it as \( \log(x+2) + \log(25) = 2 \), then combine to get \( \log(25(x+2)) = 2 \), leading to \( 25(x+2) = 100 \), so \( x = 2 \).

Front

What is the definition of a logarithm?

Back

A logarithm is the inverse operation to exponentiation, answering the question: to what exponent must a base be raised to produce a given number.

Front

What is the change of base formula for logarithms?

Back

The change of base formula states that \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \) for any positive base \( k \).

Front

What is the relationship between logarithms and exponents?

Back

Logarithms and exponents are inverses; if \( b^y = x \), then \( \log_b(x) = y \).

Front

What is an extraneous solution in the context of logarithmic equations?

Back

An extraneous solution is a solution that emerges from the algebraic manipulation of an equation but does not satisfy the original equation.

Front

What is the domain of a logarithmic function?

Back

The domain of a logarithmic function \( \log_b(x) \) is \( x > 0 \).