Algebra 2 Common Core

Published by Prentice Hall
ISBN 10: 0133186024
ISBN 13: 978-0-13318-602-4

Chapter 7 - Exponential and Logarithmic Functions - 7-5 Exponential and Logarithmic Equations - Lesson Check - Page 473: 5


Error in line 3 (dropping logs around values). Correct answer: $x=16$

Work Step by Step

Notice that in the third line, the "logs" were dropped from the equation. We can do this if the two logs have the same base. However, in our case the bases ($2$ and $3$) do not match. Thus we cannot drop the logs from the equation. Instead we proceed as follows: $\log_2 x=\log_3 9^2$ $\log_2 x=\log_3 (3^2)^2$ $\log_2 x=\log_3 3^4$ Recall the power property of logarithms (pg. 462): $\log_b{m^n}=n\log_b{m}$ Applying this property to our last equation, we get: $\log_2 x=4\log_3 3$ $\log_2 x=4\times 1$ $\log_2 x=4$ We also used the fact that $\log_{3}{3}=1$ (because $3^1=3$). Next, recall the definition of a logarithm (pg. 451): $\log_{b}{x}=y$ iff $b^y=x$ Applying this to our last equation, we get: $\log_2 x=4$ $2^4=x$ $16=x$ $x=16$
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