IntroductionExponential and logarithmic functions are great tools for modeling various real-life problems, especially those that deal with fast growth and decline rates. For example, in business, a logarithmic function can be used to calculate the amount of growth in an investment over the years. This information could then help a business owner forecast future trends in the growth of investments the business relies on to fund capital spending, debt payments, and payments to stockholders.

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4.3.1: Common Logarithms

Key Concepts• Recall that a logarithmic function is the inverse of an

exponential function in the form f(x) = a(bx) + c, where a, b, and c are constants and b is greater than 0 but not equal to 1. For an exponential function, f(x) = bx, the inverse logarithmic function would be x = logb f(x). For example, the inverse logarithmic function of the exponential function g(x) = 5x is x = log5 g(x).

• Also recall that an inverse operation is the operation that reverses the effect of another operation. Applying a logarithmic operation to both sides of an exponential equation can serve as an inverse operation when solving equations with exponents. 2

4.3.1: Common Logarithms

Key Concepts, continued• Common logarithms are often used when calculating

compound interest.

• The compound interest formula can be used to calculate the interest from the original balance (the principal) and the accrued interest. The formula is

where A is the balance, P is the initial

amount, r is the annual interest rate expressed as a decimal, t is the amount of time in years, and n is the number of compounding periods per year.

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4.3.1: Common Logarithms

Key Concepts, continued• When solving for n or t in the formula, you can apply

logarithms as the inverse operation for the exponents in the formula.

• Common logarithms are also used frequently when working with pH problems.

• The pH scale is a base-10 scale that measures the acidity or alkalinity of a solution. A solution’s pH is found using the formula pH = –log [H+], where [H+] is the concentration of hydrogen ions in the solution measured in moles per liter (abbreviated as M). Generally speaking, the lower the [H+] concentration, the more acidic the solution is. 4

4.3.1: Common Logarithms

Key Concepts, continued• For instance, neutral solutions such as pure water have

a pH of 7; pure water is neither acidic nor alkaline. Solutions that have a pH of less than 7 are acidic (such as vinegar, which has a pH of about 2.4), and solutions with a pH greater than 7 are basic or alkaline(i.e., bleach, which has a pH of about 12.6).

• The pH scale ranges from the highly acidic pH 0(1 × 100 moles per liter) to the very alkaline pH 14(1 × 10–14 moles per liter).

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4.3.1: Common Logarithms

Common Errors/Misconceptions• making substitution errors when converting exponential

functions to logarithmic functions and vice versa

• misinterpreting the parts of an exponential function

• miscalculating numbers using scientific notation

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4.3.1: Common Logarithms

Guided Practice

Example 2Roger recently inherited some money. He plans to invest his inheritance in a mutual fund earning 8.5% interest compounded annually. How long will it take for him to double his inheritance? Round your final calculation to the nearest tenth. Use the compound interest formula,

where A represents the fund’s balance,

P is the principal amount, r is the annual interest rate expressed as a decimal, n is the number of times the balance is compounded each year, and t is the time in years.

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4.3.1: Common Logarithms

Guided Practice: Example 2, continued

1. Identify the known quantities.

The annual interest rate, r, is 8.5% or, expressed as a decimal, 0.085.

The number of times the balance is compounded each year, n, is 1 because the interest is compounded annually, or one time per year.

The fund’s balance, A, is 2P because Roger’s goal is to achieve a balance that is double the principal, P.

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4.3.1: Common Logarithms

Guided Practice: Example 2, continued

2. Determine how long it will take for Roger to double his inheritance.

Use the identified values and the compound interest

formula, , to solve for t, the time.

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4.3.1: Common Logarithms

Guided Practice: Example 2, continuedCompound interest formula

Substitute 2P for A, 0.085 for r, and 1 for n.

2P = P(1.085)t Simplify.

2 = 1.085t Divide both sides by P.

log 2 = t log 1.085 Rewrite the exponential form using logarithms.

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4.3.1: Common Logarithms

Guided Practice: Example 2, continued

Divide both sides by log 1.085 and apply the

Symmetric Property of

Equality.

t ≈ 8.5 Simplify the logarithmic terms using a calculator.

It will take Roger approximately 8.5 years to

double his inheritance. 11

4.3.1: Common Logarithms

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Guided Practice: Example 2, continued

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4.3.1: Common Logarithms

Guided Practice

Example 4

The sound intensity level of a noise, expressed as a number of decibels

(dB), can be found using the formula , where I is the

intensity or “power” of the sound in watts per square meter. The

lowercase Greek letter beta, β, represents the sound intensity level. If

the sound intensity of a large orchestra was measured at 6.3 × 10–3 watts

per square meter, determine the number of decibels the orchestra

produced. Then, find the number of decibels produced by a sound

measured at a rock concert (I = 1 × 10–1) and the number of decibels

produced by an average conversation (I = 1 × 10–6). Compare all three

results.13

4.3.1: Common Logarithms

Guided Practice: Example 4, continued

1. Determine the number of decibels produced by the large orchestra.Substitute the sound intensity for the orchestra,I = 6.3 × 10–3, into the given formula and then solve.

Given formula

Substitute 6.3 × 10–3 for I.14

4.3.1: Common Logarithms

Guided Practice: Example 4, continuedβ = 10 log [6.3 × 10–3 – (–12)] Apply the rules of

exponents.

β = 10 log (6.3 × 109) Simplify the exponent.

β ≈ 98Evaluate the logarithm

using a calculator.

The orchestra produced a sound intensity level of approximately 98 decibels. 15

4.3.1: Common Logarithms

Guided Practice: Example 4, continued

2. Determine the number of decibels produced by the rock concert.Substitute the sound intensity for the rock concert,I = 1 × 10–1, into the given formula and then solve.

Given formula

Substitute 1 × 10–1 for I.

β = 10 log [1 × 10–1 – (–12)] Apply the rules of

exponents.

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4.3.1: Common Logarithms

Guided Practice: Example 4, continuedβ = 10 log (1 × 1011) Simplify the

exponent.

β = 110Evaluate the logarithm

using a calculator.

The rock concert produced a sound intensity level of 110 decibels.

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4.3.1: Common Logarithms

Guided Practice: Example 4, continued

3. Determine the number of decibels produced by the average conversation.Substitute the sound intensity for the average conversation, I = 1 × 10–6, into the given formula and then solve.

Given formula

Substitute 1 × 10–6 for I. 18

4.3.1: Common Logarithms

Guided Practice: Example 4, continuedβ = 10 log [1 × 10–6 – (–12)] Apply the rules

of exponents.

β = 10 log (1 × 106) Simplify the exponent.

β = 10 log 106

Simplify.

β = 60Evaluate the logarithm

using a calculator.

The average conversation produced a sound intensity level of 60 decibels.

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4.3.1: Common Logarithms

Guided Practice: Example 4, continued

4. Compare the three sounds by sound intensity level.The previous calculations show that the rock concert has the highest sound intensity level at 110 dB, followed by the orchestra performance at approximately 98 dB, and then the average conversation at 60 dB.

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4.3.1: Common Logarithms

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Guided Practice: Example 4, continued

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4.3.1: Common Logarithms