Solving Logarithmic Equations

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What is a logarithm?

Back

A logarithm is the exponent to which a base must be raised to produce a given number. For example, in the expression log_b(a) = c, b^c = a.

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What is the base of a logarithm?

Back

The base of a logarithm is the number that is raised to a power. In log_b(a), b is the base.

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What is the common logarithm?

Back

The common logarithm is the logarithm with base 10, denoted as log₁₀(x) or simply log(x).

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What is the natural logarithm?

Back

The natural logarithm is the logarithm with base e (approximately 2.718), denoted as ln(x) or log_e(x).

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How do you solve the equation log_b(x) = y?

Back

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

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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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What is the product rule for logarithms?

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

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What is the quotient rule for logarithms?

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

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What is the power rule for logarithms?

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The power rule states that log_b(xⁿ) = n * log_b(x).

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How do you solve log₅(4x-7) = log₅(x+5)?

Back

Set the arguments equal to each other: 4x - 7 = x + 5. Solve for x to find x = 4.