DK: Exponentials and Logarithms

Front

What is the definition of an exponential function?

Back

An exponential function is a mathematical function of the form f(x) = a * b^x, where a is a constant, b is a positive real number, and x is the variable. It describes growth or decay processes.

Front

What is the base of a logarithm?

Back

The base of a logarithm is the number that is raised to a power to obtain a given number. For example, in log_b(a) = c, b is the base.

Front

What is the relationship between exponents and logarithms?

Back

Logarithms are the inverse operations of exponentiation. If b^c = a, then log_b(a) = c.

Front

Expand the expression: 5log_5(a) - 6log_5(b)

Back

5log_5(a) - 6log_5(b) = 5log_5(a) - 30log_5(b) (using the property k*log_b(x) = log_b(x^k)).

Front

Evaluate: Log_2(64)

Back

Log_2(64) = 6, because 2^6 = 64.

Front

What is the power rule for logarithms?

Back

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

Front

Simplify the expression: (t^{-6})^{-5}

Back

(t^{-6})^{-5} = t^{30} (using the property (x^m)^n = x^{m*n}).

Front

Condense the logarithm: log_8(x^6) - log_8(y^3)

Back

log_8(x^6) - log_8(y^3) = log_8(x^6/y^3) (using the property log_b(x) - log_b(y) = log_b(x/y)).

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 product rule for logarithms?

Back

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