Exponential and Logarithmic Equations and Models

College Algebra

Product Rule for Logarithms

The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.

log$(𝑀𝑁) = log$ 𝑀 + log$ 𝑁 for 𝑏 > 0

Example: Expand log. 30𝑥 3𝑥 + 4Solution:

= log. 30𝑥 + log. 30𝑥 + 4= log. 30 + log. 𝑥 + log. 30𝑥 + 4

Quotient Rule for Logarithms

The quotient rule for logarithms can be used to simplify a logarithm of a quotient by rewriting it as the difference of individual logarithms.

log$𝑀𝑁 = log$ 𝑀 − log$ 𝑁

Example: Expand log 345674.468

Solution: = log 34(46.).(46.)

= log 34.

= log 2 + log 𝑥 − log 3

Power Rule for Logarithms

The power rule for logarithms can be used to simplify a logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.

log$ 𝑀: = 𝑛log$ 𝑀

Example: Expand log. 25Solution:

= log. 53= 2 log. 5

Expand Logarithmic Expressions Using the Logarithm Rules

Taken together, the product rule, quotient rule, and power rule are often called “laws of logs.” Sometimes we apply more than one rule in order to simplify an expression.

Example: Expand log345

=>

Solution:= log3 𝑥3 − log3 𝑦.= 2 log3 𝑥 − 3 log3 𝑦

Condense Logarithmic Expressions

Given a sum, difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm:

1. Apply the power property first. Identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power.

2. Next apply the product property. Rewrite sums of logarithms as the logarithm of a product.

3. Apply the quotient property last. Rewrite differences of logarithms as the logarithm of a quotient.

Change of Base for Logarithms

The change-of-base formula can be used to evaluate a logarithm with any base.For any positive real numbers 𝑀, 𝑏, and 𝑛, where 𝑛 ≠ 1 and 𝑏 ≠ 1,

log$ 𝑀 =log: 𝑀log: 𝑏

It follows that the change-of-base formula can be used to rewrite a logarithm with any base as the quotient of common or natural logs.

log$ 𝑀 =ln𝑀ln 𝑏 =

log𝑀log 𝑏

Desmos Interactive

Topic: change of base

https://www.desmos.com/calculator/umnz24xgl1

Exponential EquationsThe one-to-one property of exponential functions can be used to solve exponential equations.For any algebraic expressions 𝑆 and 𝑇, and any positive real number 𝑏 ≠ 1,

𝑏E = 𝑏F if and only if 𝑆 = 𝑇

Given an exponential equation with the form 𝒃𝑺 = 𝒃𝑻 , where 𝑺 and 𝑻 are algebraic expressions with an unknown, solve for the unknown:1. Use the rules of exponents to simplify, if necessary, so that the resulting

equation has the form 𝑏J = 𝑏F.2. Use the one-to-one property to set the exponents equal.3. Solve the resulting equation, 𝑆 = 𝑇, for the unknown.

Exponential Equations with a Common Base

Solve: 24KL = 234KM

Solution:24KL = 234KM The common base is 2𝑥 − 1 = 2𝑥 − 4 Use the one-to-one property𝑥 = 3 Solve for x

Exponential Equations with Unlike Bases

Solve: 8463 = 1646L

Solution:

8463 = 1646L

2. 463 = 2M 46L Write 8 and 16 as powers of 2

2.467 = 2M46M To take a power of a power, multiply exponents

3𝑥 + 6 = 4𝑥 + 4 Use the one-to-one property to set the exponents equal

𝑥 = 2 Solve for x

Use Logarithms to Solve Exponential Equations

Given an exponential equation in which a common base cannot be found, solve for the unknown.1. Apply the logarithm of both sides of the equation. Use the natural

logarithm unless one of the terms in the equation has base 10.2. Use the rules of logarithms to solve for the unknown.Example: Solve 5463 = 44

ln 5463 = ln 44 Use the natural logarithm on both sides𝑥 + 2 ln 5 = 𝑥 ln 4 Use the laws of logs𝑥 ln 5 − ln 4 = −2 ln 5 Rearrange

𝑥 ln QM= ln L

3QUse the laws of logs, then solve 𝑥 = RS T.TM

RS L.3Q

Equations Containing 𝑒

One common type of exponential equations are those with base 𝑒. When we have an equation with a base 𝑒 on either side, we can use the natural logarithm to solve it.

Given an equation of the form 𝒚 = 𝑨𝒆𝒌𝒕, solve for 𝒕.1. Divide both sides of the equation by 𝐴.2. Apply the natural logarithm of both sides of the equation.3. Divide both sides of the equation by 𝑘.

Extraneous Solutions

An extraneous solution is a solution that is correct algebraically but does not satisfy the conditions of the original equation. When the logarithm is taken on both sides of the equation, remember that the argument of the logarithm must be positive. If the number we are evaluating in a logarithm function is negative, there is no output.

Example: 𝑒34 − 𝑒4 − 56 = 0𝑒4 + 7 𝑒4 − 8 = 0 Factor the quadratic equation𝑒4 = −7 or 𝑒4 = 8 Find the zeros𝑥 = ln 8 Since 𝑥 = ln −7 is not a real number, it is an

extraneous solution

Logarithmic Equations

Use the definition of a logarithm to solve logarithmic equations.For any algebraic expression 𝑆 and real numbers 𝑏 and 𝑐 where 𝑏 > 0, 𝑏 ≠ 1

log$ 𝑆 = 𝑐 if and only if 𝑏` = 𝑆

Example: log3 2 + log3 3𝑥 − 5 = 3log3 6𝑥 − 10 = 3 Apply the product rule6𝑥 − 10 = 2. Apply the definition of a logarithm𝑥 = 3 Solve for 𝑥

Logarithmic Equations

Use the one-to-one property of logarithms to solve logarithmic equations.For any real numbers x > 0,𝑆 > 0,𝑇 > 0 and any positive real number 𝑏where 𝑏 ≠ 1,

log$ 𝑆 = log$ 𝑇 if and only if S = 𝑇Example: log 3𝑥 − 2 − log 2 = log 𝑥 + 4

log .4K33

= log 𝑥 + 4 Apply the quotient rule.4K33

= 𝑥 + 4 Apply the one-to-one property

𝑥 = 10 Solve for 𝑥

Exponential Growth

The exponential growth function can be used to model the real-world phenomenon of rapid growth.

𝑦 = 𝐴T𝑒de

where 𝐴T is equal to the value at time zero and 𝑘 is a positive constant that determines the rate of growth.This function can be used in applications involving the doubling time, the time it takes for a quantity to double from it’s initial value:

2𝐴T = 𝐴T𝑒deln 2 = 𝑘𝑡

𝑡 =ln 2𝑘

Exponential DecayThe exponential decay model is used when the quantity falls rapidly toward zero.

𝑦 = 𝐴T𝑒Kde

where 𝐴T is equal to the value at time zero and 𝑘 is a negative constant that determines the rate of decay.The half-life is the time it takes for a substance to exponentially decay to half of its original quantity.

12𝐴T = 𝐴T𝑒de

ln L3= 𝑘𝑡, or − ln 2 = 𝑘𝑡

𝑡 =−ln 2𝑘

Desmos Interactive

Topic: exponential growth and decay

https://www.desmos.com/calculator/4zae7vnjfr

Quick Review

• What are the three rules that comprise the “laws of logs”?• Can we expand ln(𝑥3 + 𝑦3)?• Can we change common logarithms to natural logarithms?• What is the one-to-one property for exponential functions?• What is an extraneous solution?• What is the half-life of an exponential decay model?• Is there any way to solve 24 = 34?• How can we solve log3 7 on a calculator that has ln and log buttons?