Summary: Exponential and Logarithmic Equations

Key Equations

One-to-one property for exponential functions For any algebraic expressions S and T and any positive real number b, where [latex]b>0,\text{ }b\ne 1, {b}^{S}={b}^{T}[/latex] if and only if S = T.
Definition of a logarithm For any algebraic expression S and positive real numbers b and c, where [latex]b\ne 1[/latex], [latex]{\mathrm{log}}_{b}\left(S\right)=c[/latex] if and only if [latex]{b}^{c}=S[/latex].
One-to-one property for logarithmic functions For any algebraic expressions S and T and any positive real number b, where [latex]b\ne 1[/latex],
[latex]{\mathrm{log}}_{b}S={\mathrm{log}}_{b}T[/latex] if and only if S = T.

Key Concepts

  • We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.
  • When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown.
  • When we are given an exponential equation where the bases are not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown.
  • When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side.
  • We can solve exponential equations with base e by applying the natural logarithm to both sides because exponential and logarithmic functions are inverses of each other.
  • After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions.
  • When given an equation of the form [latex]{\mathrm{log}}_{b}\left(S\right)=c[/latex], where S is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation [latex]{b}^{c}=S[/latex] and solve for the unknown.
  • We can also use graphing to solve equations of the form [latex]{\mathrm{log}}_{b}\left(S\right)=c[/latex]. We graph both equations [latex]y={\mathrm{log}}_{b}\left(S\right)[/latex] and [latex]y=c[/latex] on the same coordinate plane and identify the solution as the x-value of the point of intersecting.
  • When given an equation of the form [latex]{\mathrm{log}}_{b}S={\mathrm{log}}_{b}T[/latex], where S and T are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation S = T for the unknown.
  • Combining the skills learned in this and previous sections, we can solve equations that model real world situations whether the unknown is in an exponent or in the argument of a logarithm.

Glossary

extraneous solution

a solution introduced while solving an equation that does not satisfy the conditions of the original equation