User:Guy vandegrift/sandbox

Alternative proof

We begin with an equation that relates two exponentials with two different bases, $(a,b)$, both positive and neither equal to one:

$b^y=a^z$,

Randomly select and solve for one of the bases. If $b$ is selected:

$b=a^{z/y}$

Take the logarithm of both sides, randomly taking the logarithm's base to be either $a$ or $b$. If $a$ is selected:

\(\ds \log_a b\)

\(=\)

\(\ds \log_a\left( a^{z/y}\right)\)

\(\ds \)

\(=\)

\(\ds \dfrac z y\, \log_a a\)

Logarithms of Powers

\(\ds \log_a b\)

\(=\)

\(\ds \dfrac z y\)

Using $\log_a a=1$.

Now define, $x=b^y=a^z$, and note that:

$\quad b^y = x \Longrightarrow y = \log_b x,$	Definition:General Logarithm
$\quad a^z = x \Longrightarrow z = \log_a x.$

Substituting these values of $y$ and $z$ into our expression, $\log_a b=z/y$, yields the desired version of the change-of-base formula:

$\log_a b= \dfrac{\log_a x}{\log_b x}\Longrightarrow $$\boxed{\log_b x = \dfrac{\log_a x}{\log_a b}}$

$\blacksquare$

Other choices involving $a$ and $b$

This proof achieved the desired result only because two consecutive decisions were made regarding the selection of $a$ or $b$. Fortunately, there is little reason to remember these choices, provided the goal is to derive a useful formula. Suppose, for example, that we chose to solve for $b=a^{z/y}$, but instead take the logarithm of both sides at base $b$ instead of $a$. A slightly different change-of-base formula would be obtained:

$\log_b b=1=\log_b\left(a^{z/y}\right)= \dfrac{\log_a x}{\log_b x}\log_b a$$\Longrightarrow \boxed{\log_b x = (\log_b a)\cdot(\log_a x)}$

Both formulas are useful, since:

$\boxed{\left(\log_a b\right)\cdot\left(\log_b a\right)=1}$.

To understand why this identity is important, imagine that you are converting between base-2 and base-10, and have a numerical value for $\log_{10}2$, while your change-of-base formula involves $\log_2 10.$

Archives

• Special:Permalink/551517 What is physics joke (failed to parse)
• Special:Permalink/551563 and 551651 Change of Base of Logarithm