Logarithms
49 flashcards covering Logarithms for the ACT Math section.
Logarithms are a way to reverse exponentiation, helping us solve problems where a number is raised to a power. For example, if 10 raised to the power of 2 equals 100, then the logarithm base 10 of 100 is 2. Essentially, they allow us to work with exponents more easily, such as in growth models, pH calculations, or sound intensity. On the ACT, understanding logarithms is crucial because they test your ability to manipulate algebraic expressions and solve equations, which builds a strong foundation for higher math.
On the ACT Math section, logarithms appear in questions that involve solving equations, simplifying expressions, or applying properties like the product and quotient rules. Common traps include forgetting that logarithms are only defined for positive numbers or misapplying rules, such as confusing log(ab) with log(a) + log(b) incorrectly. Focus on practicing with multiple-choice formats, memorizing key properties, and double-checking your work to avoid careless errors.
A concrete tip: Always practice converting between exponential and logarithmic forms to build intuition.
Terms (49)
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Logarithm
A logarithm is the exponent to which a specified base must be raised to obtain a given number, such as logb(a) = c means b^c = a.
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Base of a logarithm
The base of a logarithm is the number that is being raised to a power in the exponential form, and it must be positive and not equal to 1.
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Common logarithm
A common logarithm is a logarithm with base 10, written as log(x), and is used to solve problems involving powers of 10.
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Natural logarithm
A natural logarithm is a logarithm with base e (approximately 2.718), written as ln(x), and is commonly used in growth and decay problems.
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Logarithmic function
A logarithmic function is a function of the form f(x) = logb(x), where b > 0 and b ≠ 1, and it is the inverse of an exponential function.
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Exponential function
An exponential function is a function of the form f(x) = b^x, where b > 0 and b ≠ 1, and its inverse is a logarithmic function.
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Inverse of exponential functions
The inverse of an exponential function y = b^x is the logarithmic function x = logb(y), which allows solving for exponents.
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Product rule of logarithms
The product rule states that logb(xy) = logb(x) + logb(y) for x > 0, y > 0, and b > 0, b ≠ 1, allowing logs of products to be simplified.
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Quotient rule of logarithms
The quotient rule states that logb(x/y) = logb(x) - logb(y) for x > 0, y > 0, and b > 0, b ≠ 1, to handle logs of quotients.
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Power rule of logarithms
The power rule states that logb(x^k) = k logb(x) for x > 0 and b > 0, b ≠ 1, which brings exponents in front of the log.
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Log of a product
For logb(xy), it equals logb(x) + logb(y), but only if x and y are positive, as this is a key property for simplifying expressions.
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Log of a quotient
For logb(x/y), it equals logb(x) - logb(y), provided x and y are positive, helping to break down division in logs.
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Log of a power
For logb(x^k), it equals k times logb(x), which is useful for rewriting expressions with exponents.
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Change of base formula
The change of base formula is logb(a) = logc(a) / logc(b) for any positive c ≠ 1, allowing calculation of logs with different bases.
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Domain of a logarithmic function
The domain of f(x) = logb(x) is x > 0, since the argument of a logarithm must be positive to be defined.
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Range of a logarithmic function
The range of f(x) = logb(x) is all real numbers, meaning logarithmic functions can output any real value.
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Graph of y = logb(x)
The graph of y = logb(x) passes through (1, 0), has a vertical asymptote at x = 0, and increases if b > 1 or decreases if 0 < b < 1.
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Vertical asymptote of logarithmic graphs
A logarithmic function y = logb(x) has a vertical asymptote at x = 0, where the function approaches negative infinity as x approaches 0 from the right.
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X-intercept of y = logb(x)
The x-intercept of y = logb(x) is at (1, 0), since logb(1) = 0 for any base b > 0, b ≠ 1.
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Increasing nature of logarithms
For b > 1, y = logb(x) is an increasing function, meaning as x increases, y increases.
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Decreasing nature of logarithms
For 0 < b < 1, y = logb(x) is a decreasing function, meaning as x increases, y decreases.
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Logarithms and exponents relationship
Logarithms and exponents are inverse operations, so if y = b^x, then x = logb(y), which is essential for solving exponential equations.
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Evaluating common logarithms
Common logarithms like log(100) = 2 because 10^2 = 100, and they are used to find exponents with base 10.
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Common mistake: log of sum
A common error is assuming logb(a + b) = logb(a) + logb(b), but it does not; the product rule applies only to products.
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Simplifying logarithmic expressions
Simplifying logarithmic expressions involves using rules like the product and power rules to combine or expand terms into a single logarithm.
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Expanding logarithmic expressions
Expanding means writing a single logarithm as a sum or difference, such as logb(xy) = logb(x) + logb(y), to break down complex logs.
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Condensing logarithmic expressions
Condensing means combining multiple logarithms into one, like logb(x) + logb(y) = logb(xy), using the product rule.
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Solving logarithmic equations
To solve equations like logb(x) = c, rewrite as x = b^c, and always check for extraneous solutions by ensuring x > 0.
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Extraneous solutions in logs
Extraneous solutions occur in logarithmic equations if they result in a non-positive argument, such as x ≤ 0, and must be discarded.
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Solving exponential equations
To solve equations like b^x = a, take the logarithm of both sides, such as x = logb(a), to isolate the exponent.
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Exponential growth model
An exponential growth model is y = a b^(kt) where b > 1, and logarithms help solve for time or other variables in growth problems.
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Exponential decay model
An exponential decay model is y = a b^(-kt) where 0 < b < 1, and logarithms are used to find decay rates or half-life.
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Half-life
Half-life is the time required for a quantity to decay to half its initial value, calculated using logarithms in exponential decay formulas.
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Doubling time
Doubling time is the time for a quantity to double in exponential growth, found by solving equations with logarithms.
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Compound interest formula
The compound interest formula is A = P(1 + r/n)^(nt), and logarithms can solve for time when the amount is given.
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Continuous compounding
Continuous compounding uses A = P e^(rt), and natural logarithms help solve for variables like time in these scenarios.
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pH scale
The pH scale measures acidity as pH = -log10([H+]), where [H+] is hydrogen ion concentration, applying logarithms to real-world chemistry.
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Richter scale
The Richter scale measures earthquake magnitude as a logarithm of the amplitude, such as M = log10(A) + adjustments, for comparing seismic events.
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Decibel scale
The decibel scale measures sound intensity as dB = 10 log10(I/I0), using logarithms to handle ratios of power.
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Logarithmic inequalities
Logarithmic inequalities, like logb(x) > c, are solved by rewriting in exponential form and considering the domain and base direction.
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Properties of logarithms with coefficients
For expressions like k logb(x), it equals logb(x^k), allowing manipulation of coefficients using the power rule.
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Base greater than 1 in logs
When the base b > 1, the logarithm function is increasing, affecting how inequalities and graphs behave.
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Base between 0 and 1 in logs
When 0 < b < 1, the logarithm function is decreasing, which reverses inequality signs in related equations.
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Common error: log(a^b) vs (log a)^b
A frequent mistake is confusing logb(a^b) = b logb(a) with (logb(a))^b, as the former is correct and the latter is not equivalent.
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Using calculators for logarithms
Calculators compute values like log(50) or ln(2) directly, but understanding the base is key for accurate application in problems.
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Natural log and e
The natural logarithm ln(x) is loge(x), where e is approximately 2.718, and it's used in formulas involving continuous change.
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e as the base
e is the base of the natural logarithm, approximately 2.718, and functions like e^x are their own derivatives, though that's beyond ACT.
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Word problems with logarithms
Word problems often involve logarithms in contexts like population growth, where you solve for time using equations like ln(P) = ln(P0) + rt.
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Identifying logarithmic patterns
Patterns in data, such as doubling times, can indicate logarithmic relationships, helping to model and solve real-world scenarios.