College Algebra (11th Edition)

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

Chapter 4 - Section 4.2 - Exponential Functions - 4.2 Exercises - Page 410: 69

Answer

$x=\frac{1}{5}$

Work Step by Step

We rewrite both sides, so that the base is $3$ on both sides. $27^{4x}=9^{x+1}$ $(3^3)^{4x}=(3^2)^{x+1}$ By the law of exponents: $(a^x)^y=a^{xy}$ Left side: $(3^3)^{4x}=3^{12x}$ Right side: $(3^2)^{x+1}=3^{2x+2}$ The rewritten equation: $3^{12x}=3^{2x+2}$ Then since the base of both sides are the same, we set the exponents equal, because the two sides are equal only if the powers are equal too: if $b^m=b^n$ and $b\ne0$, $b\ne1$ then $m=n$. $12x=2x+2 \\12x-2x=2$ $10x=2 \\\dfrac{10x}{10}=\dfrac{2}{10}$ $x=\frac{1}{5}$
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