# Logarithms

## •  Understand the definition of a logarithm as the solution to an exponential equation. •  Practice evaluating logarithmic expressions and converting between the exponential form of an equation and the logarithmic form. •  Solve exponential equations using logarithms. •  Understand the natural logarithm as a special case. •  Graph exponential and logarithmic functions, both by hand and using technology. •  (Optional) Understand and explain the properties of logarithms. •  (Optional) Use properties of logarithms to solve problems. •  (Optional) Solve problems using the properties of logarithms.

Before this unit, students can construct exponential functions given a graph, table, or description, and interpret exponential functions expressed in different forms in terms of a context. They can graph exponential functions and note key features such as end behavior, asymptotes, and intercepts. They recognize real-world contexts that can be modeled by exponential functions, such as savings accounts and population growth, and they construct functions to model them. For example, they might describe the amount in a savings account with a $1000 initial balance that earns 10% a year compounded monthly using the function$A(t) = 1000(1.0083)^{12t}$. They can solve problems involving exponential equations like$20,000 = 1000(1.0083)^{12t}\$ graphically, but not algebraically.

In this unit, students understand the logarithm defined operationally as the inverse of exponentiation, and have opportunities to practice interpreting logarithm notation and evaluating logarithms. They use logarithms to solve for an unknown exponent in situations modeled by exponential functions. These functions include ones expressed with base e, necessitating the introduction of the natural logarithm. Students graph logarithmic functions along with the exponential functions that are their inverses, developing an understanding of logarithmic functions as the inverses of exponential functions.

The Common Core State Standards do not explicitly require students to know and use the properties of logarithms. Student intending to pursue STEM careers in college should go deeper and learn these topics. Two optional sections at the end of the unit develop and apply the properties of logarithms.

After this unit, logarithms become a natural part of the toolkit in working with situations modeled by exponential functions. Applications abound in calculus, engineering, and the sciences. There are many sets of data that reveal their structure when plotted on logarithmic scales.

## Sections

M3.2.0 Pre-unit diagnostic assessment

#### Summary

Diagnose students’ ability to
• use reasoning and exponent properties to solve for an unknown exponents;
• graph a relatively simple exponential function, showing correct end behavior and intercepts;
• solve an exponential function at a given value graphically;
• summarize an exponential relation by writing an equation, given a table or several points.

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M3.2.1 Motivate the need to undo exponentiation

#### Summary

Generate a need to find an unknown exponent which is not easy to guess and check.

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M3.2.2 Understand the definition of a logarithm

#### Summary

• Understand the definition of a logarithm as the solution to an exponential equation.
• Practice evaluating log expressions and converting between the exponential form of an equation and the logarithmic form.

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M3.2.3 Use logarithms to solve problems

#### Summary

• Solve exponential equations using logarithms.
• Understand the natural logarithm as a special case.

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M3.2.4 The logarithm function

#### Summary

•  • Graph exponential and logarithmic functions, both by hand and using technology.
• Verify that f(x) = 10x and g(x) = log10(x) are inverses of one another.

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M3.2.5 Going deeper: properties of logarithms (optional)

#### Summary

• Understand and explain the properties of logarithms.
• Use properties of logarithms to solve problems.

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M3.2.6 Going deeper: using the properties of logarithms (optional)

#### Summary

Solve problems using the properties of logarithms.

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M3.2.7 Summative assessment

#### Summary

Assess students' ability to  • solve equations with unknown exponents using logarithms;
• graph a simple logarithmic function by hand showing intercepts and end behavior;
• demonstrate understanding of the inverse nature of exponential and logarithmic functions.

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