Essential Concepts

Logarithmic Properties

• We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms.
• We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms.
• We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base.
• We can use the product rule, quotient rule, and power rule together to combine or expand a logarithm with a complex input.
• The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm.
• We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula.
• The change-of-base formula is often used to rewrite a logarithm with a base other than 10 or [latex]e[/latex] as the quotient of natural or common logs. A calculator can then be used to evaluate it.

Exponential and Logarithmic Equations

• 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.
• For any algebraic expressions [latex]S[/latex] and [latex]T[/latex] and any positive real number [latex]b[/latex], where [latex]b>0,\text{ }b\ne 1, {b}^{S}={b}^{T}[/latex] if and only if [latex]S = T[/latex].
• 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 [latex]e[/latex] 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 [latex]S[/latex] 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 [latex]S[/latex] and [latex]T[/latex] are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation [latex]S = T[/latex] for the unknown.

Exponential and Logarithmic Models

• The basic exponential function is [latex]f\left(x\right)=a{b}^{x}[/latex]. If [latex]b > 1[/latex], we have exponential growth; if [latex]0 < b < 1[/latex], we have exponential decay.
• We can also write [latex]f\left(x\right)=a{b}^{x}[/latex] in terms of continuous growth as [latex]A={A}_{0}{e}^{kx}[/latex], where [latex]{A}_{0}[/latex] is the starting value. If [latex]{A}_{0}[/latex] is positive, then we have exponential growth when [latex]k > 0[/latex] and exponential decay when [latex]k < 0[/latex].
• In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict growth and decay.
• The basic exponential function is [latex]f\left(x\right)=a{b}^{x}[/latex]. If [latex]b > 1[/latex], we have exponential growth; if [latex]0 < b < 1[/latex], we have exponential decay.
• We can also write [latex]f\left(x\right)=a{b}^{x}[/latex] in terms of continuous growth as [latex]A={A}_{0}{e}^{kx}[/latex], where [latex]{A}_{0}[/latex] is the starting value. If [latex]{A}_{0}[/latex] is positive, then we have exponential growth when [latex]k > 0[/latex] and exponential decay when [latex]k < 0[/latex].
• If [latex]\text{ }A={A}_{0}{e}^{kt}[/latex], [latex]k < 0[/latex], the half-life is [latex]t=-\frac{\mathrm{ln}\left(2\right)}{k}[/latex].
• If [latex]A={A}_{0}{e}^{kt}[/latex], [latex]k > 0[/latex], the doubling time is [latex]t=\frac{\mathrm{ln}2}{k}[/latex].
• Given a substance’s doubling time or half-life, we can find a function that represents its exponential growth or decay.
• We can use Newton’s Law of Cooling ([latex]T\left(t\right)=A{e}^{kt}+{T}_{s}[/latex], where [latex]{T}_{s}[/latex] is the ambient temperature, [latex]A=T\left(0\right)-{T}_{s}[/latex], and [latex]k[/latex] is the continuous rate of cooling.) to find how long it will take for a cooling object to reach a desired temperature or to find what temperature an object will be after a given time.
• We can use logistic growth functions ([latex]f\left(x\right)=\dfrac{c}{1+a{e}^{-bx}}[/latex] where [latex]\dfrac{c}{1+a}[/latex] is the initial value, [latex]b[/latex] is a constant determined by the rate of growth, and [latex]c[/latex] is the carrying capacity or limiting value) to model real-world situations where the rate of growth changes over time, such as population growth, spread of disease, and spread of rumors.
• We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that best fit our data.
• Any exponential function of the form [latex]y=a{b}^{x}[/latex] can be rewritten as an equivalent exponential function of the form [latex]y={A}_{0}{e}^{kx}[/latex] where [latex]k=\mathrm{ln}b[/latex].
• Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without bound or where decay begins rapidly and then slows down to get closer and closer to zero.

Key Equations

The Zero Property	[latex]{\mathrm{log}}_{b}\left(1\right)=0[/latex]
The Identity Property	[latex]{\mathrm{log}}_{b}\left(b\right)=1[/latex]
The Inverse Property	[latex]{\mathrm{log}}_{b}\left(b^x\right)=x[/latex] and [latex]b^{\mathrm{log}}_{b}\left(x\right)=x[/latex]
The Product Property	[latex]{\mathrm{log}}_{b}\left(MN\right)={\mathrm{log}}_{b}\left(M\right)+{\mathrm{log}}_{b}\left(N\right)[/latex]
The Quotient Property	[latex]{\mathrm{log}}_{b}\left(\frac{M}{N}\right)={\mathrm{log}}_{b}M-{\mathrm{log}}_{b}N[/latex]
The Power Property	[latex]{\mathrm{log}}_{b}\left({M}^{n}\right)=n{\mathrm{log}}_{b}M[/latex]
The Change-of-Base Formula	[latex]{\mathrm{log}}_{b}M\text{=}\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}\text{ }n>0,n\ne 1,b\ne 1[/latex]
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.
Half-life formula	If [latex]\text{ }A={A}_{0}{e}^{kt}[/latex], k < 0, the half-life is [latex]t=-\frac{\mathrm{ln}\left(2\right)}{k}[/latex].
Carbon-14 dating	[latex]t=\frac{\mathrm{ln}\left(\frac{A}{{A}_{0}}\right)}{-0.000121}[/latex].[latex]{A}_{0}[/latex] is the amount of carbon-14 when the plant or animal died A is the amount of carbon-14 remaining today. t is that age of the fossil.
Doubling time formula	If [latex]A={A}_{0}{e}^{kt}[/latex], k > 0, the doubling time is [latex]t=\frac{\mathrm{ln}2}{k}[/latex]
Newton’s Law of Cooling	[latex]T\left(t\right)=A{e}^{kt}+{T}_{s}[/latex], where [latex]{T}_{s}[/latex] is the ambient temperature, [latex]A=T\left(0\right)-{T}_{s}[/latex], and k is the continuous rate of cooling.

Glossary

carrying capacity

in a logistic model, the limiting value of the output

change-of-base formula
a formula for converting a logarithm with any base to a quotient of logarithms with any other base

doubling time

the time it takes for a quantity to double

extraneous solution

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

half-life

the length of time it takes for a substance to exponentially decay to half of its original quantity

logistic growth model

a function of the form [latex]f\left(x\right)=\frac{c}{1+a{e}^{-bx}}[/latex] where [latex]\frac{c}{1+a}[/latex] is the initial value, c is the carrying capacity, or limiting value, and b is a constant determined by the rate of growth

Newton’s Law of Cooling

the scientific formula for temperature as a function of time as an object’s temperature is equalized with the ambient temperature

order of magnitude

the power of ten when a number is expressed in scientific notation with one non-zero digit to the left of the decimal

power rule for logarithms

a rule of logarithms that states that the log of a power is equal to the product of the exponent and the log of its base

product rule for logarithms

a rule of logarithms that states that the log of a product is equal to a sum of logarithms

quotient rule for logarithms

a rule of logarithms that states that the log of a quotient is equal to a difference of logarithms