College Algebra (11th Edition)

Published by Pearson
ISBN 10: 0321671791
ISBN 13: 978-0-32167-179-0

Chapter 4 - Section 4.3 - Logarithmic Functions - Summary Exercises on Inverse, Exponential, and Logarithmic Functions: 29

Answer

$x=-3$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given equation, $ x=\log_6 \dfrac{1}{216} ,$ use the laws of exponents and the properties of logarithms to find an equivalent expression for the logarithmic expression. $\bf{\text{Solution Details:}}$ Using exponents, the value $\dfrac{1}{216}$ is equivalent to $\dfrac{1}{6^3}.$ Hence, the equation above is equivalent to \begin{array}{l}\require{cancel} x=\log_6 \dfrac{1}{6^3} .\end{array} Using the Negative Exponent Rule of the laws of exponents which states that $x^{-m}=\dfrac{1}{x^m}$ or $\dfrac{1}{x^{-m}}=x^m,$ the equation above is equivalent to \begin{array}{l}\require{cancel} x=\log_6 6^{-3} .\end{array} Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the equation above is equivalent to \begin{array}{l}\require{cancel} x=-3\log_6 6 .\end{array} Since $\log_b b=1,$ the equation above is equivalent to \begin{array}{l}\require{cancel} x=-3(1) \\\\ x=-3 .\end{array}
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.