Logarithms

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. It is expressed as log_b(a) = c, meaning 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, allowing the logarithm to be calculated in a different base.

Front

How do you expand a logarithm?

Back

To expand a logarithm, use the properties of logarithms such as the product rule (log_b(xy) = log_b(x) + log_b(y)), the quotient rule (log_b(x/y) = log_b(x) - log_b(y)), and the power rule (log_b(x^n) = n * log_b(x)).

Front

What is the product rule of logarithms?

Back

The product rule states that the logarithm of a product is the sum of the logarithms of the factors: log_b(xy) = log_b(x) + log_b(y).

Front

What is the quotient rule of logarithms?

Back

The quotient rule states that the logarithm of a quotient is the difference of the logarithms: log_b(x/y) = log_b(x) - log_b(y).

Front

What is the power rule of logarithms?

Back

The power rule states that the logarithm of a number raised to an exponent is the exponent times the logarithm of the base: log_b(x^n) = n * log_b(x).

Front

How do you solve an exponential equation like 7^(n+10) - 8 = 6?

Back

To solve the equation, isolate the exponential term, then take the logarithm of both sides to solve for the variable n.

Front

What is the value of log_6(1/216)?

Back

Using the change of base formula, log_6(1/216) can be approximated as -3.

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How do you condense multiple logarithms into a single logarithm?

Back

To condense logarithms, use the properties of logarithms: combine sums into products, differences into quotients, and multiply exponents.

Front

What is the relationship between logarithms and exponents?

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