Question

# So far, we have always used base 10 for a logarithmic transformation. The reason for this is that our number system is based on base 10 and it is therefore easy to logarithmically transform numbers of the form . . . , 0.01, 0.1, 1, 10, 100, 1000, . . . when we use base 10. Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship in a coordinate system whose axes are accordingly transformed so that a straight line results. $y=3^{x};$ base 3

Solution

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Transform the given exponential relationship logarithmically in order to have a linear relationship.

\begin{align*} y &= 3^{x} && \text{Write exponential function}\\ \log_{3}y &= \log_{3}3^{x} && \text{Use logarithm of base 3 on both sides}\\ \log_{3}y &= x \log_{3}3 && \text{Use properties of logarithm}\\ \boldsymbol{\log_{3}y} &\boldsymbol{= x} && \text{Use properties of logarithm} \end{align*}

The linear relationship could be obtained by replacing $\log_{3}y$ with $y$.

$\boldsymbol{y = x}$

The graph of the linear relationship.

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