Logarithms and Exponentials Test Review 23-24

Front

What is the domain of the function \( f(x) = \log(x + 1) - 3 \)?

Back

The domain is \((-1, \infty)\) because the argument of the logarithm must be greater than zero.

Front

What is the value of \( e^{-3} \)?

Back

The value is approximately \( 0.0498 \).

Front

How long will it take for an investment to triple if it earns 11% interest compounded continuously?

Back

It will take approximately 9.99 years.

Front

Evaluate the logarithm: \( \log_2 32 \)

Back

The value is 5.

Front

What function models a population decreasing by 1.8% annually since 2015 if the initial population was 200,000?

Back

The function is \( y = 200,000(1 - 0.018)^t \) where \( t \) is the number of years since 2015.

Front

Define a logarithm.

Back

A logarithm is the inverse operation to exponentiation, indicating the power to which a base must be raised to obtain a given number.

Front

What is the base of the natural logarithm?

Back

The base of the natural logarithm is \( e \), approximately equal to 2.718.

Front

What is the relationship between logarithms and exponents?

Back

If \( b^y = x \), then \( \log_b(x) = y \).

Front

What is the change of base formula for logarithms?

Back

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

Front

What is the product rule for logarithms?

Back

The product rule states that \( \log_b(mn) = \log_b(m) + \log_b(n) \).