College Algebra (11th Edition)

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

Chapter 4 - Test: 14

Answer

$x\approx12.548$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given equation, $ 2^{x+1}=3^{x-4} ,$ take the logarithm of both sides. Use the properties of logarithms and of equality to isolate the variable. Approximate the answer with $3$ decimal places. $\bf{\text{Solution Details:}}$ Taking the logarithm of both sides results to \begin{array}{l}\require{cancel} \log2^{x+1}=\log3^{x-4} .\end{array} Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the equation above is equivalent \begin{array}{l}\require{cancel} (x+1)(\log2)=(x-4)(\log3) .\end{array} Using the Distributive Property and the properties of equality to isolate the variable results to \begin{array}{l}\require{cancel} x(\log2)+1(\log2)=x(\log3)-4(\log3) \\\\ x\log2+\log2=x\log3-4\log3 \\\\ x\log2-x\log3=-4\log3-\log2 \\\\ x(\log2-\log3)=-4\log3-\log2 \\\\ x=\dfrac{-4\log3-\log2}{\log2-\log3} \\\\ x\approx12.548 .\end{array}
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