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When you see expressions like 3² and 5³, you can read them as “three squared” and “five cubed.” These compact notations let you calculate the equivalent ordinary numbers—9 and 125, respectively—without expanding the multiplication.

What are Exponents?

An exponent, or power, denotes repeated multiplication of a base by itself. For example, 4⁵ = 4 × 4 × 4 × 4 × 4 = 1,024.

Special cases include any number raised to the first power remaining unchanged, and any number raised to the zero‑power equal to one: 7² = 49 and 7⁰ = 1.

Negative exponents produce reciprocals: x^-n = 1/(xⁿ). Fractional exponents represent roots; for instance, 2^5/3 means the cube root of 2 raised to the fifth power.

What are Logarithms?

Logarithms can be seen as the inverse operation of exponentiation. They answer the question: to what power must a base be raised to obtain a given number?

For example, 10³ = 1,000, which can be written as log₁₀(1,000) = 3. The general notation log_b(a) = c means that b^c = a.

Both the base and the argument must be positive, and the base cannot equal 1. When the base is omitted, it is understood to be 10 (common logarithm), while the natural logarithm uses the base e ≈ 2.7183 and is denoted ln.

Helpful Logarithm Rules

• log_b(xy) = log_b(x) + log_b(y)
• log_b(x/y) = log_b(x) – log_b(y)
• log_b(x^A) = A·log_b(x)
• log_b(1/y) = –log_b(y)

Solving for an Exponent

Consider the equation 50 = 4^x. To isolate the unknown exponent, take the logarithm of both sides (common base 10 is convenient):

log₁₀(50) = log₁₀(4^x) = x·log₁₀(4)

Thus, x = log₁₀(50) / log₁₀(4). Using a calculator, log₁₀(50) ≈ 1.699 and log₁₀(4) ≈ 0.602, giving x ≈ 2.82.

Solving Exponential Equations with e

The natural logarithm ln (base e) follows the same principles. For example, solve 16 = e^2.7x:

ln(16) = ln(e^2.7x) = 2.7x

Since ln(16) ≈ 2.773, we find x = 2.773 / 2.7 ≈ 1.03.

References

• LibreTexts Statistics: Logarithms
• Lamar University Mathematics: Solving Exponential Equations
• MathOnWeb.com: Logarithms