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Exercises Exponentials and logarithms

Example 8 from Algebra and Trigonometry, 6.7 Exponential and Logarithmic Models

Changing to base e

Change the function $y = 2.5 {(3.1)}^{x}$ so that this same function is written in the form $y = A_{0} e^{k x} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
The formula is derived as follows
$\begin{array}{l} y = 2.5 {(3.1)}^{x} \\ = 2.5 e^{\ln ({3.1}^{x})} & Insert exponential and its inverse . \\ = 2.5 e^{x \ln 3.1} & Laws of logs . \\ = 2.5 e^{(\ln 3.1)}^{x} & Commutative law of multiplication \end{array}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Change the function $y = 3 {(0.5)}^{x}$ to one having $e$ as the base.

How did it go?

PreviousChoosing a Mathematical Model Next Using Exponential Regression to Fit a Model to Data

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 6.7, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.