Solving 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

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

Back

Combine the logs: log(x(x+3)) = 1. Then, convert to exponential form: x(x+3) = 10. Solve the quadratic equation to find x = 2.

Front

What is the value of log_x(1000) = 3?

Back

This means x^3 = 1000, so x = 10.

Front

How do you solve log_8(4x+4) = 2?

Back

Convert to exponential form: 4x + 4 = 8^2 = 64. Solve for x to find x = 15.

Front

What does log_5(4x-7) = log_5(x+5) imply?

Back

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

Front

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

Back

Convert to exponential form: x + 3 = 2^4 = 16. Solve for x to find x = 13.

Front

What is the product property of logarithms?

Back

The product property states that log_b(mn) = 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).