Difficulty: Intermediate | Prerequisites: Arithmetic and Algebraic Operations.
Exponentials and logarithms are everywhere in calculus, from the chain rule to differential equations. You need to evaluate expressions like 64^(2/3) and log₂8 by hand, and know the properties of logs well enough to simplify or verify identities without a calculator.
Exponential function (b^x)
A function where the variable is in the exponent. For b > 0, b^x is defined for all real x and is always positive. The most important base in calculus is e ≈ 2.718.
Natural logarithm (ln x)
The logarithm base e. ln x is defined only for x > 0. In simple terms, ln x answers: "e raised to what power gives x?"
Logarithm (log_b x)
The inverse of the exponential: if y = b^x, then x = log_b(y). Defined for x > 0, b > 0, b ≠ 1.
Fractional exponent
b^(m/n) = the nth root of b^m. For example, 4^(3/2) = √(4³) = √64 = 8. Think of the denominator as a root and the numerator as a power.
Negative exponent
b^(-x) = 1/b^x. It flips the base into a denominator.
64^(2/3) = (64^(1/3))² = 4² = 16.
(1/16)^(-1/4) = 16^(1/4) = 2.
-8^(2/3): the negative is outside, so -(8^(2/3)) = -(8²)^(1/3) = -(64)^(1/3) = -4. A common error is to get +4.
log₂8 = 3 (because 2³ = 8).
ln 1 = 0 (because e⁰ = 1).
1/log₈2 = log₂8 = 3 (change of base: 1/log_b(a) = log_a(b)).
log 2 + log 50 = log(2 · 50) = log 100 = 2.
b^(x+y) = b^x · b^y (product rule).
(b^x)^y = b^(xy) (power rule).
b^(-x) = 1/b^x.
b^(x-y) = b^x / b^y.
x^a · x^b = x^(a+b), not x^(ab). This is a frequently tested trap.
(x^a)^b = x^(ab). True.
x^(-a) = (1/x)^a. True.
x^a + x^b ≠ x^(a+b). Addition in the base does not combine exponents.
log_b(xy) = log_b(x) + log_b(y).
log_b(x/y) = log_b(x) - log_b(y).
log_b(x^y) = y · log_b(x).
log_b(1/x) = -log_b(x).
(log b)/a = log(b^(1/a)). True.
(ln b)^a ≠ a · ln b. The power is on the value of the logarithm, not on b.
(ln a)(ln b) ≠ ln(ab). Multiplying logs is not the same as the log of a product.
3^(log₂4) = 3^2 = 9 (because log₂4 = 2).
3^(log₃6) = 6 (by the definition b^(log_b(x)) = x).
e^(ln 1) = e^0 = 1.
16^(0.5 · log₄√2) = 16^(log₄(√2)^(0.5)): log₄√2 = 1/4, so 16^(1/4 · 0.5) ... more directly, 16 = 4², so 16^(log₄√2) = (4²)^(log₄√2) = 4^(2 · log₄√2) = 4^(log₄(√2)²) = 4^(log₄2) = 2.
If x² = 2⁶, then x = 2³ = 8 (taking positive root).
The smallest integer x with 2^x ≤ 3^x is x = 0 (since 2⁰ = 3⁰ = 1).
³√(27 · 125) = ³√(3375) = 15, or equivalently ³√27 · ³√125 = 3 · 5 = 15.
e^x is defined for all real x and is never negative (always > 0).
ln x is defined only for x > 0.
e^(ln x) = x only for x > 0 (since ln x requires x > 0).
The graphs of e^x and ln x intersect: they cross near x ≈ 1.76 (so the claim they never intersect is false, but this is subtle).
log₂x can equal 0: log₂(1) = 0. So the claim it is "never equal to 0" is false.
3^π = e^(π ln 3). True (rewriting any exponential in base e).
4^π = 3^(π log₃4). True (rewriting in base 3).
Students often think (ln b)^a = a ln b. It does not. The exponent rule ln(b^a) = a ln b applies when the power is on b, not on the value of ln b.
Students often think (ln a)(ln b) = ln(ab). It does not. The product rule for logs is log a + log b = log(ab), not (log a)(log b).
Students often think if ln a < 0, then a < 0. It does not follow. ln a < 0 means 0 < a < 1 (a is positive but less than 1).
Students often confuse x^a · x^b with x^(ab). The correct rule is x^a · x^b = x^(a+b). Multiplication of like bases adds exponents; it does not multiply them.
⚠️ The derivative of e^x is e^x, and the derivative of ln x is 1/x. These are among the most-used derivative rules in the course.
⚠️ Rewriting exponentials in base e (b^x = e^(x ln b)) is needed for differentiation of general exponential functions.
⚠️ Logarithmic differentiation uses log properties to simplify products and powers before differentiating.
⚠️ If x^a + x^b = x^(a+b) appears on a true/false question, the answer is false. This is a perennial exam trap.
What is 64^(2/3)? (Answer: 16.)
What is ln 1? (Answer: 0.)
True or False: x^a · x^b = x^(a+b). (Answer: True.)
True or False: x^a · x^b = x^(ab). (Answer: False.)
True or False: e^x is never negative. (Answer: True.)
Q: Express (1/16)^(-1/4) as an integer.
A: 2. (1/16)^(-1/4) = 16^(1/4) = 2.
Q: Evaluate log₂8.
A: 3, since 2³ = 8.
Q: What is 3^(log₃6)?
A: 6. By definition, b^(log_b(x)) = x.
Q: True or False: for positive real numbers a and b, log a + log b = log(ab).
A: True. This is the product rule of logarithms.
Q: True or False: (ln b)^a = a ln b.
A: False. The correct rule is ln(b^a) = a ln b, where the exponent is on b, not on ln b.
Q: True or False: if ln a < 0, then a < 0.
A: False. ln a < 0 means 0 < a < 1.
Q: Solve for α: 2^(log₄2) · 4^(log₂4) = 2^(α/2).
A: α = 9. log₄2 = 1/2, so 2^(1/2). log₂4 = 2, so 4² = 16 = 2⁴. Product is 2^(1/2) · 2⁴ = 2^(9/2). So α/2 = 9/2, giving α = 9.
Q: True or False: x^a + x^b = x^(a+b).
A: False. Addition in the base does not combine exponents. Only x^a · x^b = x^(a+b).
The exponential function e^x and the natural logarithm ln x are central to calculus. The derivative rules d/dx(e^x) = e^x and d/dx(ln x) = 1/x appear constantly. Logarithmic differentiation is the standard technique for differentiating complicated products and powers. Exponential growth and decay models (y = Ce^(kt)) are a major application of differential equations.
Related Terms / Search Tags
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