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Logarithm Formulas and Change of Base

Review logarithm definitions, change-of-base, inverse-log, product, quotient, and power formulas with variable restrictions and calculation steps.

How do the Log Calculator formulas work?

This formula page defines the variables used by the Log Calculator, shows how those values are combined, and explains the order of operations behind the result. Use the worked steps to verify an estimate, compare scenarios, and understand which assumptions or rounding rules can affect the displayed answer.

Formula Overview

A logarithm and an exponential are inverse operations. Change of base evaluates any valid real base, while logarithm laws rewrite products, quotients, and powers.

Variable Definitions

  • b is the positive base, with b not equal to 1.
  • x is the positive logarithm argument.
  • y is the logarithm result or exponential exponent.
  • M and N are positive real expressions.
  • r is a real exponent where the expression is defined.

Logarithm Definition

y=log⁡b(x)  ⟺  by=xy=\log_b(x)\iff b^y=x

For real logarithms, x must be greater than 0, b must be greater than 0, and b cannot equal 1.

Change-of-Base Formula

log⁡b(x)=ln⁡(x)ln⁡(b)=log⁡10(x)log⁡10(b)\log_b(x)=\frac{\ln(x)}{\ln(b)}=\frac{\log_{10}(x)}{\log_{10}(b)}

This identity evaluates any valid real base using natural or common logarithms.

Inverse Logarithm

x=by=exp⁡ ⁣(yln⁡(b))x=b^y=\exp\!\left(y\ln(b)\right)

The inverse operation raises the base to the entered exponent and returns a positive value.

Product, Quotient, and Power Rules

log⁡b(MN)=log⁡b(M)+log⁡b(N)\log_b(MN)=\log_b(M)+\log_b(N)
log⁡b ⁣(MN)=log⁡b(M)−log⁡b(N)\log_b\!\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)
log⁡b(Mr)=rlog⁡b(M)\log_b(M^r)=r\log_b(M)

Graph Relationship

f(x)=log⁡b(x),f−1(x)=bxf(x)=\log_b(x),\qquad f^{-1}(x)=b^x

The graphs are reflections across y = x.

Worked Calculation

For log base 3 of 81, change of base gives ln(81) ÷ ln(3) = 4. The inverse check is 3⁴ = 81.

Formula Assumptions

  • The base is positive and not equal to 1.
  • The argument of a real logarithm is positive.
  • Product and quotient rule arguments remain positive.
  • Complex logarithms are outside scope.
  • Displayed results are rounded only after calculation.

Next Steps

Apply these formulas in the Log Calculator or verify them with the worked logarithm examples.

Use the Log Calculator