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_b(x) = c?

Back

To solve log_b(x) = c, rewrite it in exponential form: x = b^c.

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).

Front

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

Back

Rewrite in exponential form: x + 3 = 2^4, so x = 16 - 3 = 13.

Front

How do you solve log_x(1000) = 3?

Back

Rewrite in exponential form: 1000 = x^3, so x = 10.

Front

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

Back

Set the arguments equal: 4x - 7 = x + 5, solve for x to get x = 4.

Front

What is the solution to log_2(x + 2) + log_2(x) = 3?

Back

Use the product property: log_2((x + 2)x) = 3, then (x + 2)x = 2^3, solve for x to get x = 2.