{"id":289,"date":"2023-06-21T13:22:52","date_gmt":"2023-06-21T13:22:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-exponential-and-logarithmic-equations\/"},"modified":"2023-07-03T21:08:40","modified_gmt":"2023-07-03T21:08:40","slug":"summary-exponential-and-logarithmic-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-exponential-and-logarithmic-equations\/","title":{"raw":"Summary: Exponential and Logarithmic Equations","rendered":"Summary: Exponential and Logarithmic Equations"},"content":{"raw":"\n\n
Key Equations<\/h2>\n
\n\n
\n
One-to-one property for exponential functions<\/td>\n
For any algebraic expressions S<\/em> and T<\/em> and any positive real number b<\/em>, where [latex]b>0,\\text{ }b\\ne 1, {b}^{S}={b}^{T}[\/latex] if and only if S<\/em> = T<\/em>.<\/td>\n<\/tr>\n
\n
Definition of a logarithm<\/td>\n
For any algebraic expression S<\/em> and positive real numbers b<\/em> and c<\/em>, where [latex]b\\ne 1[\/latex], [latex]{\\mathrm{log}}_{b}\\left(S\\right)=c[\/latex] if and only if [latex]{b}^{c}=S[\/latex].<\/td>\n<\/tr>\n
\n
One-to-one property for logarithmic functions<\/td>\n
For any algebraic expressions S<\/em> and T<\/em> and any positive real number b<\/em>, where [latex]b\\ne 1[\/latex],\n[latex]{\\mathrm{log}}_{b}S={\\mathrm{log}}_{b}T[\/latex] if and only if S<\/em> = T<\/em>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
Key Concepts<\/h2>\n
\n \t
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.<\/li>\n \t
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.<\/li>\n \t
When we are given an exponential equation where the bases are not<\/em> 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.<\/li>\n \t
When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side.<\/li>\n \t
We can solve exponential equations with base e<\/em> by applying the natural logarithm to both sides because exponential and logarithmic functions are inverses of each other.<\/li>\n \t
After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions.<\/li>\n \t
When given an equation of the form [latex]{\\mathrm{log}}_{b}\\left(S\\right)=c[\/latex], where S<\/em> 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.<\/li>\n \t
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-<\/em>value of the point of intersecting.<\/li>\n \t
When given an equation of the form [latex]{\\mathrm{log}}_{b}S={\\mathrm{log}}_{b}T[\/latex], where S<\/em> and T<\/em> are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation S <\/em>= T<\/em> for the unknown.<\/li>\n \t
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.<\/li>\n<\/ul>\n
Glossary<\/h2>\nextraneous solution<\/strong>\n
\n \t
a solution introduced while solving an equation that does not satisfy the conditions of the original equation<\/dd>\n<\/dl>\n\n","rendered":"
Key Equations<\/h2>\n
\n\n
\n
One-to-one property for exponential functions<\/td>\n
For any algebraic expressions S<\/em> and T<\/em> and any positive real number b<\/em>, where [latex]b>0,\\text{ }b\\ne 1, {b}^{S}={b}^{T}[\/latex] if and only if S<\/em> = T<\/em>.<\/td>\n<\/tr>\n
\n
Definition of a logarithm<\/td>\n
For any algebraic expression S<\/em> and positive real numbers b<\/em> and c<\/em>, where [latex]b\\ne 1[\/latex], [latex]{\\mathrm{log}}_{b}\\left(S\\right)=c[\/latex] if and only if [latex]{b}^{c}=S[\/latex].<\/td>\n<\/tr>\n
\n
One-to-one property for logarithmic functions<\/td>\n
For any algebraic expressions S<\/em> and T<\/em> and any positive real number b<\/em>, where [latex]b\\ne 1[\/latex], \n[latex]{\\mathrm{log}}_{b}S={\\mathrm{log}}_{b}T[\/latex] if and only if S<\/em> = T<\/em>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
Key Concepts<\/h2>\n
\n
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.<\/li>\n
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.<\/li>\n
When we are given an exponential equation where the bases are not<\/em> 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.<\/li>\n
When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side.<\/li>\n
We can solve exponential equations with base e<\/em> by applying the natural logarithm to both sides because exponential and logarithmic functions are inverses of each other.<\/li>\n
After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions.<\/li>\n
When given an equation of the form [latex]{\\mathrm{log}}_{b}\\left(S\\right)=c[\/latex], where S<\/em> 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.<\/li>\n
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-<\/em>value of the point of intersecting.<\/li>\n
When given an equation of the form [latex]{\\mathrm{log}}_{b}S={\\mathrm{log}}_{b}T[\/latex], where S<\/em> and T<\/em> are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation S <\/em>= T<\/em> for the unknown.<\/li>\n
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.<\/li>\n<\/ul>\n
Glossary<\/h2>\n
extraneous solution<\/strong><\/p>\n
\n
a solution introduced while solving an equation that does not satisfy the conditions of the original equation<\/dd>\n<\/dl>\n\n\t\t\t \n\t\t\t