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The Natural Base e

The Natural Base e 1.4 - 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. In our case, since we have the number ee as a base, we will take the natural logarithm of each side. m=nln(m)=ln(n)\begin{gathered} m=n \quad \Leftrightarrow \quad \ln (m) = \ln (n) \end{gathered} Note that in order to take logarithms, both mm and nn must be positive numbers. Let's now solve our equation.
3e4x+9=153e^{4x}+9=15
3e4x=63e^{4x}=6
e4x=2e^{4x}=2
ln(LHS)=ln(RHS)\ln(\text{LHS})=\ln(\text{RHS})
lne4x=ln(2)\ln e^{4x}= \ln (2)
ln(ea)=a \ln\left(e^a\right) = a
4x=ln(2)4x= \ln (2)
Calculate logarithm
4x=0.69314718064x=0.6931471806\ldots
x=0.1732867951x=0.1732867951\ldots
x0.173x\approx 0.173