Vocabulary
Exponential Equation- An equation in which a variable expression occurs as an exponent.
Logarithmic equation- An equation that involves a logarithm of a variable expression.
Extraneous solution- An apparent solution that must be rejected because it does not satisfy the original equation.
Logarithmic equation- An equation that involves a logarithm of a variable expression.
Extraneous solution- An apparent solution that must be rejected because it does not satisfy the original equation.
Essential Information
Exponential equations are equations in which variable expressions occur as exponents.
Property of Equality for Exponential Equations
If b is a positive number other than 1, then bⁿ = b⁵ if and only if n=5.
Property of Equality for Exponential Equations
If b is a positive number other than 1, then bⁿ = b⁵ if and only if n=5.
When it is not convenient to write each side of an exponential equation using the same base, the equation can be solved by taking the logarithm of each side.
Newtons's Law of Cooling
An important application of exponential equations is Newton's law of cooling. this law states that for a cooling substance with initial temperature T₀, the temperature T after t minutes can be modeled by
T = (T₀ -Tᵣ)e^(-Rt) +Tᵣ
where Tᵣ is the surrounding temperature and R is the substance's cooling rate.
An important application of exponential equations is Newton's law of cooling. this law states that for a cooling substance with initial temperature T₀, the temperature T after t minutes can be modeled by
T = (T₀ -Tᵣ)e^(-Rt) +Tᵣ
where Tᵣ is the surrounding temperature and R is the substance's cooling rate.
Solving Logarithmic Equations
Logarithmic equations are equations that involve logarithms of variable expressions. the following property can be used to solve some types of logarithmic equations.
If b, x, and y are positive numbers with b≠1, then logᵦx=logᵦy if and only if x=y.
Logarithmic equations are equations that involve logarithms of variable expressions. the following property can be used to solve some types of logarithmic equations.
If b, x, and y are positive numbers with b≠1, then logᵦx=logᵦy if and only if x=y.
Exponentiating to Solve EquationsThe property of equality for equations implies that given an equation x=y, then exponentiate each side to obtain an equation of the form b^x=b^y. This technique is useful for solving some logarithmic equations.
Extraneous Solutions
Because the domain of a logarithmic function generally does not include all real numbers, be sure to check for extraneous solutions of logarithmic equations. This can be done algebraically or graphically.
Because the domain of a logarithmic function generally does not include all real numbers, be sure to check for extraneous solutions of logarithmic equations. This can be done algebraically or graphically.