Chapter 12/13.2

Front

What is the property of logarithms that allows you to combine logarithms with the same base?

Back

The properties of logarithms state that \( \log_a b + \log_a c = \log_a (bc) \) and \( \log_a b - \log_a c = \log_a \left( \frac{b}{c} \right) \).

Front

Convert the logarithmic equation \( \log_{2}8=3 \) to exponential form.

Back

The exponential form is \( 2^3=8 \).

Front

What is the definition of a one-to-one function?

Back

A function is one-to-one if it never assigns the same value to two different domain elements, meaning if \( f(a) = f(b) \), then \( a = b \).

Front

How do you express \( 3\log_a6-3\log_a2 \) using properties of logarithms?

Back

This can be expressed as \( 3(\log_a 6 - \log_a 2) = 3\log_a \left( \frac{6}{2} \right) = 3\log_a 3 = \log_a 27 \).

Front

What is the graph of the function \( f(x)=3^{3x}-1 \) as x approaches infinity?

Back

As \( x \) approaches infinity, \( f(x) \) approaches infinity as well, indicating an exponential growth.

Front

What is the logarithmic identity for \( \log_a a \)?

Back

The identity states that \( \log_a a = 1 \) for any base \( a \) where \( a > 0 \).

Front

What is the change of base formula for logarithms?

Back

The change of base formula is given by \( \log_a b = \frac{\log_c b}{\log_c a} \) for any positive base \( c \).

Front

How do you simplify \( \log_x\left(\frac{c^4f^3}{k^8}\right) \) using properties of logarithms?

Back

It simplifies to \( \log_x c^4 + \log_x f^3 - \log_x k^8 = 4\log_x c + 3\log_x f - 8\log_x k \).

Front

What does it mean for a function to be constant?

Back

A constant function is one that does not change its value; for example, \( h(x) = 2 \) is constant and is not one-to-one.

Front

What is the relationship between logarithms and exponents?

Back

Logarithms are the inverse operations of exponentiation; for example, if \( b^y = x \), then \( \log_b x = y \).