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Using Properties of Logarithms

Using Properties of Logarithms 1.5 - Solution

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When bases are not the same, we can solve an exponential equation by taking logarithms on each side of the equation. m=nlog(m)=log(n)\begin{gathered} m=n \quad \Leftrightarrow \quad \log (m) = \log (n) \end{gathered} Note that in order to take logarithms, both mm and nn must be positive numbers. Let's now solve our equation.
116x=3811^{6x}=38
log(LHS)=log(RHS)\log_{}(\text{LHS})=\log_{}(\text{RHS})
log(116x)=log(38)\log \left( 11^{6x} \right)= \log (38)
log(am)=mlog(a)\log_{}\left(a^m\right)= m\cdot \log_{}(a)
6xlog(11)=log(38)6x \cdot \log (11)= \log (38)
6x=log(38)log(11)6x=\dfrac{\log (38)}{\log (11)}
6x=1.5169912556x= 1.516991255\ldots
x=0.2528318759x=0.2528318759\ldots
x0.253x\approx 0.253