# Motivate the need to undo exponentiation

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

The goal of this section is to help students see why a logarithm might be useful in their mathematical toolkit. Students are presented with a situation where they develop an exponential model and need to find an unknown exponent that yields a specified value. Teachers can revisit and assess solving such a problem by graphing with technology or by guess and check. They can then point out that students can solve all other kinds of equations that they know about by rewriting them in a helpful, equivalent form (for example, $x^3 = 1000$ can be rewritten as $x = 1000^{1/3}$) and suggest that there should be a way to do that to find an unknown exponent (as in, for example, $3^x = 1000$). Teachers can either introduce the logarithm at this point, or leave students in suspense until the next section.

## External Resources

1 I Remember

#### Description

WHAT: Students listen to a Radiolab episode which explains that every time we remember something, we alter the memory slightly. They explore the fidelity of a memory as a function of the number of remembrances, comparing decreasing linear and exponential decay models (F-BF.A.1$^\star$, MP.4), discussing which is more realistic, and exploring the implications of memory deterioration. They determine the number of remembrances (the exponent) that corresponds to given fidelity threshold, thus motivating the need for a logarithm (F-LE.A.4$^\star$).

WHY: The purpose of this activity is to active studentsâ€™ prior knowledge of linear and exponential functions and to highlight the contrast between the simplicity of solving a linear equation algebraically with the need to use technology to solve an exponential equation. This raises the possibility that there might be a new operation that solves exponential equations.

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