## 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