3.4 - Solving Logarithmic Equations

Front

Define a logarithm.

Back

A logarithm is the exponent to which a base must be raised to produce a given number. For example, log_b(a) = c means b^c = a.

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 is the product property of logarithms?

Back

The product property states that log_b(m * n) = log_b(m) + log_b(n).

Front

What is the quotient property of logarithms?

Back

The quotient property states that log_b(m / n) = log_b(m) - log_b(n).

Front

What is the power property of logarithms?

Back

The power property states that log_b(m^p) = p * log_b(m).

Front

How do you solve the equation log(x) + log(x+3) = 1?

Back

Combine the logs: log(x(x+3)) = 1. Then, x(x+3) = 10, leading to the quadratic equation x^2 + 3x - 10 = 0. The solution is x = 2.

Front

How do you solve the equation log_5(4x - 7) = log_5(x + 5)?

Back

Since the logs are equal, set the arguments equal: 4x - 7 = x + 5. Solve for x to get x = 4.

Front

How do you solve the equation log_4(3x - 1) = log_4(2x + 3)?

Back

Set the arguments equal: 3x - 1 = 2x + 3. Solve for x to get x = 4.

Front

How do you solve the equation log_2(x + 3) = 4?

Back

Convert to exponential form: x + 3 = 2^4. Solve for x to get x = 13.

Front

How do you solve the equation log(10x) + log(x - 3) = 2?

Back

Combine the logs: log(10x(x - 3)) = 2. Convert to exponential form: 10x(x - 3) = 100. Solve for x to get x = 5.