SOLVE EXPONENTIAL AND LOGARITHMIC EQUATIONS

Front

What is an exponential equation?

Back

An exponential equation is an equation in which a variable appears in the exponent. For example, 4^(3x + 5) = 16.

Front

How do you solve the equation 4^(3x + 5) = 16?

Back

Convert 16 to a power of 4: 4^(3x + 5) = 4^2. Therefore, 3x + 5 = 2, leading to x = -1.

Front

What is a logarithmic equation?

Back

A logarithmic equation is an equation that involves a logarithm of a variable. For example, log_2(2 - 2x) + log_2(1 - x) = 5.

Front

How do you solve log_2(2 - 2x) + log_2(1 - x) = 5?

Back

Combine the logs: log_2((2 - 2x)(1 - x)) = 5. This implies (2 - 2x)(1 - x) = 32, leading to x = -3.

Front

What is the change of base formula for logarithms?

Back

The change of base formula states that log_b(a) = log_k(a) / log_k(b) for any positive k.

Front

What does ln(x) represent?

Back

ln(x) represents the natural logarithm of x, which is the logarithm to the base e (approximately 2.718).

Front

How do you solve ln(x - 2) - ln(3x) = 0?

Back

Combine the logs: ln((x - 2)/(3x)) = 0. This implies (x - 2)/(3x) = 1, leading to no solution.

Front

What is the domain of a logarithmic function?

Back

The domain of a logarithmic function is the set of positive real numbers, as the logarithm of zero or a negative number is undefined.

Front

What is the relationship between exponential and logarithmic functions?

Back

Exponential functions and logarithmic functions are inverses of each other. If y = b^x, then x = log_b(y).

Front

How do you solve the equation 2(16^(x+6)) - 8 = 35?

Back

First, isolate the exponential term: 2(16^(x+6)) = 43. Then, solve for x, leading to x = -4.89.