Logarithmic Equations and Properties

Front

What is a logarithm?

Back

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

Front

What is the base of a logarithm?

Back

The base of a logarithm is the number that is raised to a power. Common bases are 10 (common logarithm) and e (natural logarithm).

Front

What is the product property of logarithms?

Back

The product property states that log_b(xy) = log_b(x) + log_b(y).

Front

What is the quotient property of logarithms?

Back

The quotient property states that log_b(x/y) = log_b(x) - log_b(y).

Front

What is the power property of logarithms?

Back

The power property states that log_b(x^k) = k * log_b(x).

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

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.

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How do you express log(xy^2) using logarithmic properties?

Back

log(xy^2) = log(x) + log(y^2) = log(x) + 2*log(y).

Front

What is the definition of a logarithmic equation?

Back

A logarithmic equation is an equation that involves a logarithm with a variable argument.

Front

How do you solve log_2(x^3) = 5?

Back

Rewrite in exponential form: x^3 = 2^5. Then, x = (2^5)^(1/3) = 4.