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Calculus.3 What Theorem 6.7 really tells us is that all exponential and logarithmic functions are
just scalings of one another. Not only does this explain why their graphs have similar shapes, but
it also tells us that we could do all of mathematics with a single base - be it 10, e, 42, or 117. Your
Calculus teacher will have more to say about this when the time comes.
Example 6.2.3. Use an appropriate change of base formula to convert the following expressions to
ones with the indicated base. Verify your answers using a calculator, as appropriate.
1. 32 to base 10 3. log4 (5) to base e 2. 2x to base e 4. ln(x) to base 10 Solution.
1. We apply the Change of Base formula with a = 3 and b = 10 to obtain 32 = 102 log(3) . Typing
the latter in the calculator produces an answer of 9 as required.
2. Here, a = 2 and b = e so we have 2x = ex ln(2) . To verify this on our calculator, we can graph
f (x) = 2x and g (x) = ex ln(2) . Their graphs are indistinguishable which provides evidence
that they are the same function. y = f (x) = 2x and y = g (x) = ex ln(2)
3
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