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Exponents and Logarithms — Definition, Formula & Examples

Exponents and logarithms are inverse operations that describe repeated multiplication and its reverse. An exponent tells you how many times to multiply a base by itself, while a logarithm asks what exponent produces a given result.

If b>0b > 0, b≠1b \neq 1, and bn=xb^n = x where x>0x > 0, then nn is the exponent and log⁡bx=n\log_b x = n is its corresponding logarithm. The exponential function f(x)=bxf(x) = b^x and the logarithmic function g(x)=log⁡bxg(x) = \log_b x are inverses of each other, meaning blog⁡bx=xb^{\log_b x} = x and log⁡b(bx)=x\log_b(b^x) = x.

Key Formula

\text{Exponent Laws:}\quad b^m \cdot b^n = b^{m+n}, \quad \frac{b^m}{b^n} = b^{m-n}, \quad (b^m)^n = b^{mn}$$ $$\text{Log Rules:}\quad \log_b(xy) = \log_b x + \log_b y, \quad \log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y, \quad \log_b(x^n) = n\log_b x
Where:
  • bb = The base of the exponent or logarithm (b > 0, b ≠ 1)
  • m,nm, n = Exponents (any real numbers for the exponent laws)
  • x,yx, y = Positive real numbers (arguments of the logarithm)

How It Works

Exponent laws let you simplify expressions involving powers of the same base: you add exponents when multiplying, subtract when dividing, and multiply exponents when raising a power to a power. Logarithm rules mirror these laws — the log of a product becomes a sum, the log of a quotient becomes a difference, and the log of a power brings the exponent out front. To solve an exponential equation like 3x=813^x = 81, you can rewrite it using logarithms: x=log⁡381x = \log_3 81. To solve a logarithmic equation like log⁡2x=5\log_2 x = 5, you convert to exponential form: x=25=32x = 2^5 = 32. These two operations always undo each other, which is the key idea connecting them.

Worked Example

Problem: Solve for x: 52x=6255^{2x} = 625.
Step 1: Recognize that 625 is a power of 5.
625=54625 = 5^4
Step 2: Rewrite the equation so both sides have the same base.
52x=545^{2x} = 5^4
Step 3: Since the bases match, set the exponents equal.
2x=42x = 4
Step 4: Solve for x.
x=2x = 2
Answer: x=2x = 2

Another Example

Problem: Use logarithm rules to expand log⁡2(8x3)\log_2(8x^3).
Step 1: Apply the product rule to separate the two factors.
log⁡2(8x3)=log⁡28+log⁡2(x3)\log_2(8x^3) = \log_2 8 + \log_2(x^3)
Step 2: Evaluate the first term: since 23=82^3 = 8, we get log⁡28=3\log_2 8 = 3.
=3+log⁡2(x3)= 3 + \log_2(x^3)
Step 3: Apply the power rule to bring the exponent out front.
=3+3log⁡2x= 3 + 3\log_2 x
Answer: log⁡2(8x3)=3+3log⁡2x\log_2(8x^3) = 3 + 3\log_2 x

Visualization

Why It Matters

Exponents and logarithms appear throughout Algebra 2, Precalculus, and AP Calculus when you model population growth, radioactive decay, compound interest, and sound intensity (decibels). Scientists use logarithmic scales like the Richter scale and pH scale precisely because logarithms compress huge ranges of values into manageable numbers. Mastering these rules is also essential for any college course in engineering, biology, or finance that involves exponential models.

Common Mistakes

Mistake: Adding exponents when the bases are different, such as writing 23⋅32=652^3 \cdot 3^2 = 6^5.
Correction: The product rule bm⋅bn=bm+nb^m \cdot b^n = b^{m+n} only works when the bases are identical. If the bases differ, you must evaluate each power separately: 23⋅32=8⋅9=722^3 \cdot 3^2 = 8 \cdot 9 = 72.
Mistake: Treating log⁡b(x+y)\log_b(x + y) as log⁡bx+log⁡by\log_b x + \log_b y.
Correction: There is no log rule for the log of a sum. The product rule says log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y. The expression log⁡b(x+y)\log_b(x + y) cannot be simplified further with basic log rules.

Related Terms