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Chapter 6

Exponential and Logarithmic Functions

Exponential and logarithmic functions go together. You wouldn’t think so at first glance, because exponential functions can look like , and logarithmic (log) functions can look like . What joins them together is that exponential functions and log functions are inverses of each other.

Exponential and logarithmic functions can have bases that are any positive number except the number 1. The special cases are those with base 10 (common logarithms) and base e (natural logarithms), which go along with their exponential counterparts.

The whole point of these functions is to tell you how large something is when you use a particular exponent or how big of an exponent you need in order to create a particular number. These functions are heavily used in the sciences and finance, so studying them here can pay off big time in later studies.

The Problems You’ll Work On

In this chapter, you’ll work with exponential and logarithmic functions in the following ways:

 Evaluating exponential and log functions using the function rule

 Simplifying expressions involving exponential and log functions

 Solving exponential equations using rules involving exponents

 Solving logarithmic equations using laws of logarithms

 Graphing exponential and logarithmic functions for a better view of their powers

 Applying exponential and logarithmic functions to real-life situations

What to Watch Out For

Don’t let common mistakes trip you up. Here are some of the challenges you’ll face when working with exponential and logarithmic functions:

 Using the rules for exponents in various operations correctly

 Applying the laws of logarithms to denominators of fractions

 Remembering the order of operations when simplifying exponential and log expressions

 Checking for extraneous roots when solving logarithmic equations

Understanding Function Notation

321–325 Evaluate the function at the indicated points.

321. Evaluate the function at and .

322. Evaluate the function at and .

323. Evaluate the function at and .

324. Evaluate the function at and .

325. Evaluate the function at and .

Graphing Exponential Functions

326–330 Graph the exponential function.

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Solving Exponential Equations

331–345 Solve the exponential equation for x.

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Using the Equivalence to Rewrite Expressions

346–348 Rewrite the exponential expression as a logarithmic expression.

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Using the Equivalence to Rewrite Expressions

349–350 Rewrite the logarithmic expression as an exponential expression.

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Rewriting Logarithmic Expressions

351–358 Write the expression in a new form using the laws of logarithms.

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Rewriting Logs of Products and Quotients as Sums and Differences

359–365 Write the expression as an expanded logarithm.

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Solving Logarithmic Equations

366–375 Solve the logarithmic equation for x.

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Applying Function Transformations to Log Functions

376–380 Find the best choice for a function rule of the transformation of shown in the graph.

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Illustration by Thomson Digital

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Illustration by Thomson Digital

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Illustration by Thomson Digital

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Illustration by Thomson Digital

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Illustration by Thomson Digital

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Applying Logarithms to Everyday Life

381–390 Use the given formula and properties of logarithms to answer the question.

381. Orange juice has a hydrogen ion concentration of 0.001 moles/liter. What is the pH of orange juice? Use the pH formula , where H+ is the hydrogen ion concentration in moles/liter.

382. Ammonia water has a pH of 11.6. What is its hydrogen ion concentration (in moles/liter)? Use the pH formula , where H+ is the hydrogen ion concentration in moles/liter.

383. You invest $5,000 in an account at an annual rate of 1.475% with interest compounding quarterly. Find out how much is in the account (to the nearest cent) after 3 years by using the compound interest formula , where A is the total amount in the account in dollars, P is the principal (amount invested), r is the annual interest rate, n is the number of times compounded per year, and t is the time in years.

384. If you invest $1,000 at a rate of 3.25%, compounded continuously, how many years would it take for your money to double? Round to the nearest hundredth of a year. Use the continuous compound interest formula , where A is the amount in the account, P is the principal, r is the interest rate, and t is the time in years.

385. You buy a car that costs $23,495, and it loses $4,500 in value when you drive it off the car lot. The car then loses 10.785% in value each year (exponential depreciation). Rounded to the nearest cent, what is the value of the car after 5 years? Use the exponential equation , where y is the resulting value, C is the initial amount, k is the growth or decay rate, and t is the time in years.

386. A normal conversation is 60 decibels (dB). What is the intensity of the sound of the conversation (in watts per square meter)? Use the following formula, where L is decibels and I is the intensity of the sound: .

387. Two earthquakes occurred in different parts of the country. The first earthquake had a magnitude of 8.0, and the second had a magnitude of 4.0. Compare the intensity of the earthquakes using the equation , where M is the magnitude, I is the intensity measured from the epicenter, and I₀ is a constant.

388. A town had a population of 6,250 in 1975 and a population of 8,125 in 2010. If the rate of growth is exponential, what will the population be in 2040? Use the exponential model , where y is the resulting population, C is the initial amount, k is the growth or decay rate, and t is the time in years.

389. In 2000, a house was appraised at $179,900. In 2013, it was reappraised at $138,000. If the decline rate is exponential, what will the value of the house be in 2020? Round the value to the nearest dollar. Use the exponential model , where y is the resulting value, C is the initial amount, k is the growth or decay rate, and t is the time in years.

390. One student with a contagious flu virus goes to school when they’re sick. The school district includes 7,500 students and 1,000 staff members. The growth of the virus is modeled by the following equation, where y is the number infected and t is time in days: . When 45% of the students and staff members fall ill, the schools will close. How many days before the schools are closed? Round to the nearest whole day.