3.3 Solve logarithmic equations

Front

What is the definition of 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(4x) - log(4) = log(12)?

Back

Combine the logs using the quotient property: log(4x/4) = log(12). This simplifies to log(x) = log(12), so x = 12.

Front

How do you solve the equation log(2x) - log(5) = log(30)?

Back

Combine the logs using the quotient property: log(2x/5) = log(30). This simplifies to 2x/5 = 30, so x = 75.

Front

How do you solve the equation log(x+6) - log(x) = log(20)?

Back

Combine the logs using the quotient property: log((x+6)/x) = log(20). This simplifies to (x+6)/x = 20, so x = 6/19.

Front

How do you solve the equation log_4(x) = 3?

Back

Rewrite in exponential form: x = 4^3 = 64.

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. Solving gives x = 4.