College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 4, Exponential and Logarithmic Functions - Section 4.5 - Exponential and Logarithmic Functions - 4.5 Exercises - Page 404: 32


(a) $x=\frac{\ln\frac{100}{3}}{3\ln 5}$ (b) $x\approx 0.726249$

Work Step by Step

(a) We need to solve: $125^{x}+5^{3x+1}=200 $ $(5^3)^x+5^{3x+1}=200$ $5^{3x}(1+5^1)=200$ $5^{3x}(6)=200$ $5^{3x}=\frac{200}{6}=\frac{100}{3}$ We take the log of both sides and use log rules to simplify: $\ln 5^{3x}=\ln \frac{100}{3}$ $3x\ln 5=\ln 100-\ln 3=\ln \frac{100}{3}$ $x=\frac{\ln\frac{100}{3}}{3\ln 5}$ (b) We solve using a calculator: $x\approx 0.726249$
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